Authors Robert A. Beezer
License GFDL-1.2-no-invariants-or-later
Flashcard Supplement
to
A First Course in Linear Algebra
Robert A. Beezer
University of Puget Sound
Version 3.00-preview(December 30, 2015)
Robert A. Beezer is a Professor of Mathematics at the University of Puget Sound, where he has
been on the faculty since 1984. He received a B.S. in Mathematics (with an Emphasis in Computer
Science) from the University of Santa Clara in 1978, a M.S. in Statistics from the University of
Illinois at Urbana-Champaign in 1982 and a Ph.D. in Mathematics from the University of Illinois at
Urbana-Champaign in 1984.
In addition to his teaching at the University of Puget Sound, he has made sabbatical visits to
the University of the West Indies (Trinidad campus) and the University of Western Australia. He
has also given several courses in the Master’s program at the African Institute for Mathematical
Sciences, South Africa. He has been a Sage developer since 2008.
He teaches calculus, linear algebra and abstract algebra regularly, while his research inter-
ests include the applications of linear algebra to graph theory. His professional website is at
http://buzzard.ups.edu.
Edition
Version 3.50 Flashcard Supplement
December 30, 2015
Publisher
Robert A. Beezer Congruent Press Gig Harbor, Washington, USA
c 2004—2015 Robert A. Beezer
Permission is granted to copy, distribute and/or modify this document under the terms of the
GNU Free Documentation License, Version 1.2 or any later version published by the Free Software
Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of
the license is included in the appendix entitled “GNU Free Documentation License”.
Definition SLE System of Linear Equations 1
A system of linear equations is a collection of m equations in the variable quantities
x1 , x2 , x3 , . . . , xn of the form,
a11 x1 + a12 x2 + a13 x3 + · · · + a1n xn = b1
a21 x1 + a22 x2 + a23 x3 + · · · + a2n xn = b2
a31 x1 + a32 x2 + a33 x3 + · · · + a3n xn = b3
..
.
am1 x1 + am2 x2 + am3 x3 + · · · + amn xn = bm
where the values of aij , bi and xj , 1 ≤ i ≤ m, 1 ≤ j ≤ n, are from the set of complex numbers,
C.
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Definition SSLE Solution of a System of Linear Equations 2
A solution of a system of linear equations in n variables, x1 , x2 , x3 , . . . , xn (such as the system
given in Definition SLE), is an ordered list of n complex numbers, s1 , s2 , s3 , . . . , sn such that if
we substitute s1 for x1 , s2 for x2 , s3 for x3 , , sn for xn , then for every equation of the system
the left side will equal the right side, i.e. each equation is true simultaneously.
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Definition SSSLE Solution Set of a System of Linear Equations 3
The solution set of a linear system of equations is the set which contains every solution to the
system, and nothing more.
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Definition ESYS Equivalent Systems 4
Two systems of linear equations are equivalent if their solution sets are equal.
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Definition EO Equation Operations 5
Given a system of linear equations, the following three operations will transform the system into
a different one, and each operation is known as an equation operation.
1. Swap the locations of two equations in the list of equations.
2. Multiply each term of an equation by a nonzero quantity.
3. Multiply each term of one equation by some quantity, and add these terms to a second equa-
tion, on both sides of the equality. Leave the first equation the same after this operation,
but replace the second equation by the new one.
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Theorem EOPSS Equation Operations Preserve Solution Sets 6
If we apply one of the three equation operations of Definition EO to a system of linear equations
(Definition SLE), then the original system and the transformed system are equivalent.
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Definition M Matrix 7
An m × n matrix is a rectangular layout of numbers from C having m rows and n columns. We
will use upper-case Latin letters from the start of the alphabet (A, B, C, . . . ) to denote matrices
and squared-off brackets to delimit the layout. Many use large parentheses instead of brackets
— the distinction is not important. Rows of a matrix will be referenced starting at the top and
working down (i.e. row 1 is at the top) and columns will be referenced starting from the left (i.e.
column 1 is at the left). For a matrix A, the notation [A]ij will refer to the complex number in
row i and column j of A.
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Definition CV Column Vector 8
A column vector of size m is an ordered list of m numbers, which is written in order vertically,
starting at the top and proceeding to the bottom. At times, we will refer to a column vector
as simply a vector. Column vectors will be written in bold, usually with lower case Latin letter
from the end of the alphabet such as u, v, w, x, y, z. Some books like to write vectors with
arrows, such as ~u. Writing by hand, some like to put arrows on top of the symbol, or a tilde
underneath the symbol, as in u. To refer to the entry or component of vector v in location i of
∼
the list, we write [v]i .
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Definition ZCV Zero Column Vector 9
The zero vector of size m is the column vector of size m where each entry is the number zero,
0
0
0 = 0
..
.
0
or defined much more compactly, [0]i = 0 for 1 ≤ i ≤ m.
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Definition CM Coefficient Matrix 10
For a system of linear equations,
a11 x1 + a12 x2 + a13 x3 + · · · + a1n xn = b1
a21 x1 + a22 x2 + a23 x3 + · · · + a2n xn = b2
a31 x1 + a32 x2 + a33 x3 + · · · + a3n xn = b3
..
.
am1 x1 + am2 x2 + am3 x3 + · · · + amn xn = bm
the coefficient matrix is the m × n matrix
a11 a12 a13 ... a1n
a21 a22 a23 ... a2n
A = a31 a32 a33 ... a3n
..
.
am1 am2 am3 ... amn
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Definition VOC Vector of Constants 11
For a system of linear equations,
a11 x1 + a12 x2 + a13 x3 + · · · + a1n xn = b1
a21 x1 + a22 x2 + a23 x3 + · · · + a2n xn = b2
a31 x1 + a32 x2 + a33 x3 + · · · + a3n xn = b3
..
.
am1 x1 + am2 x2 + am3 x3 + · · · + amn xn = bm
the vector of constants is the column vector of size m
b1
b2
b = b3
..
.
bm
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Definition SOLV Solution Vector 12
For a system of linear equations,
a11 x1 + a12 x2 + a13 x3 + · · · + a1n xn = b1
a21 x1 + a22 x2 + a23 x3 + · · · + a2n xn = b2
a31 x1 + a32 x2 + a33 x3 + · · · + a3n xn = b3
..
.
am1 x1 + am2 x2 + am3 x3 + · · · + amn xn = bm
the solution vector is the column vector of size n
x1
x2
x = x3
..
.
xn
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Definition MRLS Matrix Representation of a Linear System 13
If A is the coefficient matrix of a system of linear equations and b is the vector of constants,
then we will write LS(A, b) as a shorthand expression for the system of linear equations, which
we will refer to as the matrix representation of the linear system.
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Definition AM Augmented Matrix 14
Suppose we have a system of m equations in n variables, with coefficient matrix A and vector of
constants b. Then the augmented matrix of the system of equations is the m × (n + 1) matrix
whose first n columns are the columns of A and whose last column (n + 1) is the column vector
b. This matrix will be written as [ A | b].
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Definition RO Row Operations 15
The following three operations will transform an m × n matrix into a different matrix of the same
size, and each is known as a row operation.
1. Swap the locations of two rows.
2. Multiply each entry of a single row by a nonzero quantity.
3. Multiply each entry of one row by some quantity, and add these values to the entries in
the same columns of a second row. Leave the first row the same after this operation, but
replace the second row by the new values.
We will use a symbolic shorthand to describe these row operations:
1. Ri ↔ Rj : Swap the location of rows i and j.
2. αRi : Multiply row i by the nonzero scalar α.
3. αRi + Rj : Multiply row i by the scalar α and add to row j.
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Definition REM Row-Equivalent Matrices 16
Two matrices, A and B, are row-equivalent if one can be obtained from the other by a sequence
of row operations.
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Theorem REMES Row-Equivalent Matrices represent Equivalent Systems 17
Suppose that A and B are row-equivalent augmented matrices. Then the systems of linear
equations that they represent are equivalent systems.
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Definition RREF Reduced Row-Echelon Form 18
A matrix is in reduced row-echelon form if it meets all of the following conditions:
1. If there is a row where every entry is zero, then this row lies below any other row that
contains a nonzero entry.
2. The leftmost nonzero entry of a row is equal to 1.
3. The leftmost nonzero entry of a row is the only nonzero entry in its column.
4. Consider any two different leftmost nonzero entries, one located in row i, column j and the
other located in row s, column t. If s > i, then t > j.
A row of only zero entries is called a zero row and the leftmost nonzero entry of a nonzero row
is a leading 1. A column containing a leading 1 will be called a pivot column. The number
of nonzero rows will be denoted by r, which is also equal to the number of leading 1’s and the
number of pivot columns.
The set of column indices for the pivot columns will be denoted by D = {d1 , d2 , d3 , . . . , dr }
where d1 < d2 < d3 < · · · < dr , while the columns that are not pivot columns will be denoted as
F = {f1 , f2 , f3 , . . . , fn−r } where f1 < f2 < f3 < · · · < fn−r .
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Theorem REMEF Row-Equivalent Matrix in Echelon Form 19
Suppose A is a matrix. Then there is a matrix B so that
1. A and B are row-equivalent.
2. B is in reduced row-echelon form.
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Theorem RREFU Reduced Row-Echelon Form is Unique 20
Suppose that A is an m × n matrix and that B and C are m × n matrices that are row-equivalent
to A and in reduced row-echelon form. Then B = C.
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Definition CS Consistent System 21
A system of linear equations is consistent if it has at least one solution. Otherwise, the system
is called inconsistent.
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Definition IDV Independent and Dependent Variables 22
Suppose A is the augmented matrix of a consistent system of linear equations and B is a row-
equivalent matrix in reduced row-echelon form. Suppose j is the index of a pivot column of B.
Then the variable xj is dependent. A variable that is not dependent is called independent or
free.
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Theorem RCLS Recognizing Consistency of a Linear System 23
Suppose A is the augmented matrix of a system of linear equations with n variables. Suppose
also that B is a row-equivalent matrix in reduced row-echelon form with r nonzero rows. Then
the system of equations is inconsistent if and only if column n + 1 of B is a pivot column.
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Theorem CSRN Consistent Systems, r and n 24
Suppose A is the augmented matrix of a consistent system of linear equations with n variables.
Suppose also that B is a row-equivalent matrix in reduced row-echelon form with r pivot columns.
Then r ≤ n. If r = n, then the system has a unique solution, and if r < n, then the system has
infinitely many solutions.
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Theorem FVCS Free Variables for Consistent Systems 25
Suppose A is the augmented matrix of a consistent system of linear equations with n variables.
Suppose also that B is a row-equivalent matrix in reduced row-echelon form with r rows that are
not completely zeros. Then the solution set can be described with n − r free variables.
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Theorem PSSLS Possible Solution Sets for Linear Systems 26
A system of linear equations has no solutions, a unique solution or infinitely many solutions.
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Theorem CMVEI Consistent, More Variables than Equations, Infinite solutions27
Suppose a consistent system of linear equations has m equations in n variables. If n > m, then
the system has infinitely many solutions.
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Definition HS Homogeneous System 28
A system of linear equations, LS(A, b) is homogeneous if the vector of constants is the zero
vector, in other words, if b = 0.
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Theorem HSC Homogeneous Systems are Consistent 29
Suppose that a system of linear equations is homogeneous. Then the system is consistent and
one solution is found by setting each variable to zero.
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Definition TSHSE Trivial Solution to Homogeneous Systems of Equations 30
Suppose a homogeneous system of linear equations has n variables. The solution x1 = 0, x2 = 0,
, xn = 0 (i.e. x = 0) is called the trivial solution.
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Theorem HMVEI Homogeneous, More Variables than Equations, Infinite solutions
31
Suppose that a homogeneous system of linear equations has m equations and n variables with
n > m. Then the system has infinitely many solutions.
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Definition NSM Null Space of a Matrix 32
The null space of a matrix A, denoted N (A), is the set of all the vectors that are solutions to
the homogeneous system LS(A, 0).
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Definition SQM Square Matrix 33
A matrix with m rows and n columns is square if m = n. In this case, we say the matrix has
size n. To emphasize the situation when a matrix is not square, we will call it rectangular.
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Definition NM Nonsingular Matrix 34
Suppose A is a square matrix. Suppose further that the solution set to the homogeneous linear
system of equations LS(A, 0) is {0}, in other words, the system has only the trivial solution.
Then we say that A is a nonsingular matrix. Otherwise we say A is a singular matrix.
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Definition IM Identity Matrix 35
The m × m identity matrix, Im , is defined by
(
1 i=j
[Im ]ij = 1 ≤ i, j ≤ m
0 i 6= j
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Theorem NMRRI Nonsingular Matrices Row Reduce to the Identity matrix 36
Suppose that A is a square matrix and B is a row-equivalent matrix in reduced row-echelon form.
Then A is nonsingular if and only if B is the identity matrix.
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Theorem NMTNS Nonsingular Matrices have Trivial Null Spaces 37
Suppose that A is a square matrix. Then A is nonsingular if and only if the null space of A is
the set containing only the zero vector, i.e. N (A) = {0}.
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Theorem NMUS Nonsingular Matrices and Unique Solutions 38
Suppose that A is a square matrix. A is a nonsingular matrix if and only if the system LS(A, b)
has a unique solution for every choice of the constant vector b.
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Theorem NME1 Nonsingular Matrix Equivalences, Round 1 39
Suppose that A is a square matrix. The following are equivalent.
1. A is nonsingular.
2. A row-reduces to the identity matrix.
3. The null space of A contains only the zero vector, N (A) = {0}.
4. The linear system LS(A, b) has a unique solution for every possible choice of b.
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Definition VSCV Vector Space of Column Vectors 40
The vector space Cm is the set of all column vectors (Definition CV) of size m with entries from
the set of complex numbers, C.
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Definition CVE Column Vector Equality 41
Suppose that u, v ∈ Cm . Then u and v are equal, written u = v if
[u]i = [v]i 1≤i≤m
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Definition CVA Column Vector Addition 42
Suppose that u, v ∈ Cm . The sum of u and v is the vector u + v defined by
[u + v]i = [u]i + [v]i 1≤i≤m
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Definition CVSM Column Vector Scalar Multiplication 43
Suppose u ∈ Cm and α ∈ C, then the scalar multiple of u by α is the vector αu defined by
[αu]i = α [u]i 1≤i≤m
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Theorem VSPCV Vector Space Properties of Column Vectors 44
Suppose that Cm is the set of column vectors of size m (Definition VSCV) with addition and
scalar multiplication as defined in Definition CVA and Definition CVSM. Then
• ACC Additive Closure, Column Vectors: If u, v ∈ Cm , then u + v ∈ Cm .
• SCC Scalar Closure, Column Vectors: If α ∈ C and u ∈ Cm , then αu ∈ Cm .
• CC Commutativity, Column Vectors: If u, v ∈ Cm , then u + v = v + u.
• AAC Additive Associativity, Column Vectors: If u, v, w ∈ Cm , then u + (v + w) =
(u + v) + w.
• ZC Zero Vector, Column Vectors: There is a vector, 0, called the zero vector, such that
u + 0 = u for all u ∈ Cm .
• AIC Additive Inverses, Column Vectors: If u ∈ Cm , then there exists a vector −u ∈ Cm so
that u + (−u) = 0.
• SMAC Scalar Multiplication Associativity, Column Vectors: If α, β ∈ C and u ∈ Cm , then
α(βu) = (αβ)u.
• DVAC Distributivity across Vector Addition, Column Vectors: If α ∈ C and u, v ∈ Cm ,
then α(u + v) = αu + αv.
• DSAC Distributivity across Scalar Addition, Column Vectors: If α, β ∈ C and u ∈ Cm ,
then (α + β)u = αu + βu.
• OC One, Column Vectors: If u ∈ Cm , then 1u = u.
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Definition LCCV Linear Combination of Column Vectors 45
Given n vectors u1 , u2 , u3 , . . . , un from Cm and n scalars α1 , α2 , α3 , . . . , αn , their linear com-
bination is the vector
α1 u1 + α2 u2 + α3 u3 + · · · + αn un
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Theorem SLSLC Solutions to Linear Systems are Linear Combinations 46
Denote the columns of the m×n matrix A as the vectors A1 , A2 , A3 , . . . , An . Then x ∈ Cn is a
solution to the linear system of equations LS(A, b) if and only if b equals the linear combination
of the columns of A formed with the entries of x,
[x]1 A1 + [x]2 A2 + [x]3 A3 + · · · + [x]n An = b
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Theorem VFSLS Vector Form of Solutions to Linear Systems 47
Suppose that [ A | b] is the augmented matrix for a consistent linear system LS(A, b) of m
equations in n variables. Let B be a row-equivalent m × (n + 1) matrix in reduced row-echelon
form. Suppose that B has r pivot columns, with indices D = {d1 , d2 , d3 , . . . , dr }, while the
n − r non-pivot columns have indices in F = {f1 , f2 , f3 , . . . , fn−r , n + 1}. Define vectors c, uj ,
1 ≤ j ≤ n − r of size n by
(
0 if i ∈ F
[c]i =
[B]k,n+1 if i ∈ D, i = dk
1
if i ∈ F , i = fj
[uj ]i = 0 if i ∈ F , i 6= fj .
− [B]
k,fj if i ∈ D, i = dk
Then the set of solutions to the system of equations LS(A, b) is
S = { c + α1 u1 + α2 u2 + α3 u3 + · · · + αn−r un−r | α1 , α2 , α3 , . . . , αn−r ∈ C}
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Theorem PSPHS Particular Solution Plus Homogeneous Solutions 48
Suppose that w is one solution to the linear system of equations LS(A, b). Then y is a solution
to LS(A, b) if and only if y = w + z for some vector z ∈ N (A).
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Definition SSCV Span of a Set of Column Vectors 49
Given a set of vectors S = {u1 , u2 , u3 , . . . , up }, their span, hSi, is the set of all possible linear
combinations of u1 , u2 , u3 , . . . , up . Symbolically,
hSi = { α1 u1 + α2 u2 + α3 u3 + · · · + αp up | αi ∈ C, 1 ≤ i ≤ p}
( p )
X
= αi ui αi ∈ C, 1 ≤ i ≤ p
i=1
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Theorem SSNS Spanning Sets for Null Spaces 50
Suppose that A is an m × n matrix, and B is a row-equivalent matrix in reduced row-echelon
form. Suppose that B has r pivot columns, with indices given by D = {d1 , d2 , d3 , . . . , dr }, while
the n − r non-pivot columns have indices F = {f1 , f2 , f3 , . . . , fn−r , n + 1}. Construct the n − r
vectors zj , 1 ≤ j ≤ n − r of size n,
1
if i ∈ F , i = fj
[zj ]i = 0 if i ∈ F , i 6= fj
− [B]
k,fj if i ∈ D, i = dk
Then the null space of A is given by
N (A) = h{z1 , z2 , z3 , . . . , zn−r }i
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Definition RLDCV Relation of Linear Dependence for Column Vectors 51
Given a set of vectors S = {u1 , u2 , u3 , . . . , un }, a true statement of the form
α1 u1 + α2 u2 + α3 u3 + · · · + αn un = 0
is a relation of linear dependence on S. If this statement is formed in a trivial fashion, i.e. αi = 0,
1 ≤ i ≤ n, then we say it is the trivial relation of linear dependence on S.
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Definition LICV Linear Independence of Column Vectors 52
The set of vectors S = {u1 , u2 , u3 , . . . , un } is linearly dependent if there is a relation of linear
dependence on S that is not trivial. In the case where the only relation of linear dependence on
S is the trivial one, then S is a linearly independent set of vectors.
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Theorem LIVHS Linearly Independent Vectors and Homogeneous Systems 53
Suppose that S = {v1 , v2 , v3 , . . . , vn } ⊆ Cm is a set of vectors and A is the m × n matrix whose
columns are the vectors in S. Then S is a linearly independent set if and only if the homogeneous
system LS(A, 0) has a unique solution.
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Theorem LIVRN Linearly Independent Vectors, r and n 54
Suppose that S = {v1 , v2 , v3 , . . . , vn } ⊆ Cm is a set of vectors and A is the m × n matrix
whose columns are the vectors in S. Let B be a matrix in reduced row-echelon form that is
row-equivalent to A and let r denote the number of pivot columns in B. Then S is linearly
independent if and only if n = r.
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Theorem MVSLD More Vectors than Size implies Linear Dependence 55
Suppose that S = {u1 , u2 , u3 , . . . , un } ⊆ Cm and n > m. Then S is a linearly dependent set.
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Theorem NMLIC Nonsingular Matrices have Linearly Independent Columns 56
Suppose that A is a square matrix. Then A is nonsingular if and only if the columns of A form
a linearly independent set.
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Theorem NME2 Nonsingular Matrix Equivalences, Round 2 57
Suppose that A is a square matrix. The following are equivalent.
1. A is nonsingular.
2. A row-reduces to the identity matrix.
3. The null space of A contains only the zero vector, N (A) = {0}.
4. The linear system LS(A, b) has a unique solution for every possible choice of b.
5. The columns of A form a linearly independent set.
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Theorem BNS Basis for Null Spaces 58
Suppose that A is an m × n matrix, and B is a row-equivalent matrix in reduced row-echelon
form with r pivot columns. Let D = {d1 , d2 , d3 , . . . , dr } and F = {f1 , f2 , f3 , . . . , fn−r } be the
sets of column indices where B does and does not (respectively) have pivot columns. Construct
the n − r vectors zj , 1 ≤ j ≤ n − r of size n as
1
if i ∈ F , i = fj
[zj ]i = 0 if i ∈ F , i 6= fj
− [B]
k,fj if i ∈ D, i = dk
Define the set S = {z1 , z2 , z3 , . . . , zn−r }.Then
1. N (A) = hSi.
2. S is a linearly independent set.
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Theorem DLDS Dependency in Linearly Dependent Sets 59
Suppose that S = {u1 , u2 , u3 , . . . , un } is a set of vectors. Then S is a linearly dependent set
if and only if there is an index t, 1 ≤ t ≤ n such that ut is a linear combination of the vectors
u1 , u2 , u3 , . . . , ut−1 , ut+1 , . . . , un .
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Theorem BS Basis of a Span 60
Suppose that S = {v1 , v2 , v3 , . . . , vn } is a set of column vectors. Define W = hSi and let A be
the matrix whose columns are the vectors from S. Let B be the reduced row-echelon form of A,
with D = {d1 , d2 , d3 , . . . , dr } the set of indices for the pivot columns of B. Then
1. T = {vd1 , vd2 , vd3 , . . . vdr } is a linearly independent set.
2. W = hT i.
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Definition CCCV Complex Conjugate of a Column Vector 61
Suppose that u is a vector from Cm . Then the conjugate of the vector, u, is defined by
[u]i = [u]i 1≤i≤m
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Theorem CRVA Conjugation Respects Vector Addition 62
Suppose x and y are two vectors from Cm . Then
x+y =x+y
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Theorem CRSM Conjugation Respects Vector Scalar Multiplication 63
Suppose x is a vector from Cm , and α ∈ C is a scalar. Then
αx = α x
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Definition IP Inner Product 64
Given the vectors u, v ∈ Cm the inner product of u and v is the scalar quantity in C,
m
X
hu, vi = [u]1 [v]1 + [u]2 [v]2 + [u]3 [v]3 + · · · + [u]m [v]m = [u]i [v]i
i=1
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Theorem IPVA Inner Product and Vector Addition 65
Suppose u, v, w ∈ Cm . Then
1. hu + v, wi = hu, wi + hv, wi
2. hu, v + wi = hu, vi + hu, wi
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Theorem IPSM Inner Product and Scalar Multiplication 66
Suppose u, v ∈ Cm and α ∈ C. Then
1. hαu, vi = α hu, vi
2. hu, αvi = α hu, vi
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Theorem IPAC Inner Product is Anti-Commutative 67
Suppose that u and v are vectors in Cm . Then hu, vi = hv, ui.
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Definition NV Norm of a Vector 68
The norm of the vector u is the scalar quantity in C
v
q um
2 2 2 2
uX 2
kuk = |[u]1 | + |[u]2 | + |[u]3 | + · · · + |[u]m | = t |[u]i |
i=1
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Theorem IPN Inner Products and Norms 69
2
Suppose that u is a vector in Cm . Then kuk = hu, ui.
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Theorem PIP Positive Inner Products 70
Suppose that u is a vector in Cm . Then hu, ui ≥ 0 with equality if and only if u = 0.
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Definition OV Orthogonal Vectors 71
A pair of vectors, u and v, from Cm are orthogonal if their inner product is zero, that is,
hu, vi = 0.
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Definition OSV Orthogonal Set of Vectors 72
Suppose that S = {u1 , u2 , u3 , . . . , un } is a set of vectors from Cm . Then S is an orthogonal set
if every pair of different vectors from S is orthogonal, that is hui , uj i = 0 whenever i 6= j.
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Definition SUV Standard Unit Vectors 73
Let ej ∈ Cm , 1 ≤ j ≤ m denote the column vectors defined by
(
0 if i 6= j
[ej ]i =
1 if i = j
Then the set
{e1 , e2 , e3 , . . . , em } = { ej | 1 ≤ j ≤ m}
is the set of standard unit vectors in Cm .
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Theorem OSLI Orthogonal Sets are Linearly Independent 74
Suppose that S is an orthogonal set of nonzero vectors. Then S is linearly independent.
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Theorem GSP Gram-Schmidt Procedure 75
Suppose that S = {v1 , v2 , v3 , . . . , vp } is a linearly independent set of vectors in Cm . Define
the vectors ui , 1 ≤ i ≤ p by
hu1 , vi i hu2 , vi i hu3 , vi i hui−1 , vi i
ui = vi − u1 − u2 − u3 − · · · − ui−1
hu1 , u1 i hu2 , u2 i hu3 , u3 i hui−1 , ui−1 i
Let T = {u1 , u2 , u3 , . . . , up }. Then T is an orthogonal set of nonzero vectors, and hT i = hSi.
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Definition ONS OrthoNormal Set 76
Suppose S = {u1 , u2 , u3 , . . . , un } is an orthogonal set of vectors such that kui k = 1 for all
1 ≤ i ≤ n. Then S is an orthonormal set of vectors.
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Definition VSM Vector Space of m × n Matrices 77
The vector space Mmn is the set of all m × n matrices with entries from the set of complex
numbers.
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Definition ME Matrix Equality 78
The m × n matrices A and B are equal, written A = B provided [A]ij = [B]ij for all 1 ≤ i ≤ m,
1 ≤ j ≤ n.
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Definition MA Matrix Addition 79
Given the m × n matrices A and B, define the sum of A and B as an m × n matrix, written
A + B, according to
[A + B]ij = [A]ij + [B]ij 1 ≤ i ≤ m, 1 ≤ j ≤ n
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Definition MSM Matrix Scalar Multiplication 80
Given the m × n matrix A and the scalar α ∈ C, the scalar multiple of A is an m × n matrix,
written αA and defined according to
[αA]ij = α [A]ij 1 ≤ i ≤ m, 1 ≤ j ≤ n
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Theorem VSPM Vector Space Properties of Matrices 81
Suppose that Mmn is the set of all m × n matrices (Definition VSM) with addition and scalar
multiplication as defined in Definition MA and Definition MSM. Then
• ACM Additive Closure, Matrices: If A, B ∈ Mmn , then A + B ∈ Mmn .
• SCM Scalar Closure, Matrices: If α ∈ C and A ∈ Mmn , then αA ∈ Mmn .
• CM Commutativity, Matrices: If A, B ∈ Mmn , then A + B = B + A.
• AAM Additive Associativity, Matrices: If A, B, C ∈ Mmn , then A + (B + C) = (A + B) +
C.
• ZM Zero Matrix, Matrices: There is a matrix, O, called the zero matrix, such that A + O =
A for all A ∈ Mmn .
• AIM Additive Inverses, Matrices: If A ∈ Mmn , then there exists a matrix −A ∈ Mmn so
that A + (−A) = O.
• SMAM Scalar Multiplication Associativity, Matrices: If α, β ∈ C and A ∈ Mmn , then
α(βA) = (αβ)A.
• DMAM Distributivity across Matrix Addition, Matrices: If α ∈ C and A, B ∈ Mmn , then
α(A + B) = αA + αB.
• DSAM Distributivity across Scalar Addition, Matrices: If α, β ∈ C and A ∈ Mmn , then
(α + β)A = αA + βA.
• OM One, Matrices: If A ∈ Mmn , then 1A = A.
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Definition ZM Zero Matrix 82
The m × n zero matrix is written as O = Om×n and defined by [O]ij = 0, for all 1 ≤ i ≤ m,
1 ≤ j ≤ n.
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Definition TM Transpose of a Matrix 83
Given an m × n matrix A, its transpose is the n × m matrix At given by
t
A ij = [A]ji , 1 ≤ i ≤ n, 1 ≤ j ≤ m.
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Definition SYM Symmetric Matrix 84
The matrix A is symmetric if A = At .
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Theorem SMS Symmetric Matrices are Square 85
Suppose that A is a symmetric matrix. Then A is square.
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Theorem TMA Transpose and Matrix Addition 86
Suppose that A and B are m × n matrices. Then (A + B)t = At + B t .
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Theorem TMSM Transpose and Matrix Scalar Multiplication 87
Suppose that α ∈ C and A is an m × n matrix. Then (αA)t = αAt .
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Theorem TT Transpose of a Transpose 88
t
Suppose that A is an m × n matrix. Then (At ) = A.
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Definition CCM Complex Conjugate of a Matrix 89
Suppose A is an m × n matrix. Then the conjugate of A, written A is an m × n matrix defined
by
A ij = [A]ij
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Theorem CRMA Conjugation Respects Matrix Addition 90
Suppose that A and B are m × n matrices. Then A + B = A + B.
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Theorem CRMSM Conjugation Respects Matrix Scalar Multiplication 91
Suppose that α ∈ C and A is an m × n matrix. Then αA = αA.
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Theorem CCM Conjugate of the Conjugate of a Matrix 92
Suppose that A is an m × n matrix. Then A = A.
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Theorem MCT Matrix Conjugation and Transposes 93
t
Suppose that A is an m × n matrix. Then (At ) = A .
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Definition A Adjoint 94
t
If A is a matrix, then its adjoint is A∗ = A .
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Theorem AMA Adjoint and Matrix Addition 95
∗
Suppose A and B are matrices of the same size. Then (A + B) = A∗ + B ∗ .
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Theorem AMSM Adjoint and Matrix Scalar Multiplication 96
∗
Suppose α ∈ C is a scalar and A is a matrix. Then (αA) = αA∗ .
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Theorem AA Adjoint of an Adjoint 97
∗
Suppose that A is a matrix. Then (A∗ ) = A.
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Definition MVP Matrix-Vector Product 98
Suppose A is an m × n matrix with columns A1 , A2 , A3 , . . . , An and u is a vector of size n.
Then the matrix-vector product of A with u is the linear combination
Au = [u]1 A1 + [u]2 A2 + [u]3 A3 + · · · + [u]n An
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Theorem SLEMM Systems of Linear Equations as Matrix Multiplication 99
The set of solutions to the linear system LS(A, b) equals the set of solutions for x in the vector
equation Ax = b.
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Theorem EMMVP Equal Matrices and Matrix-Vector Products 100
Suppose that A and B are m × n matrices such that Ax = Bx for every x ∈ Cn . Then A = B.
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Definition MM Matrix Multiplication 101
Suppose A is an m × n matrix and B1 , B2 , B3 , . . . , Bp are the columns of an n × p matrix B.
Then the matrix product of A with B is the m × p matrix where column i is the matrix-vector
product ABi . Symbolically,
AB = A [B1 |B2 |B3 | . . . |Bp ] = [AB1 |AB2 |AB3 | . . . |ABp ] .
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Theorem EMP Entries of Matrix Products 102
Suppose A is an m × n matrix and B is an n × p matrix. Then for 1 ≤ i ≤ m, 1 ≤ j ≤ p, the
individual entries of AB are given by
[AB]ij = [A]i1 [B]1j + [A]i2 [B]2j + [A]i3 [B]3j + · · · + [A]in [B]nj
n
X
= [A]ik [B]kj
k=1
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Theorem MMZM Matrix Multiplication and the Zero Matrix 103
Suppose A is an m × n matrix. Then
1. AOn×p = Om×p
2. Op×m A = Op×n
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Theorem MMIM Matrix Multiplication and Identity Matrix 104
Suppose A is an m × n matrix. Then
1. AIn = A
2. Im A = A
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Theorem MMDAA Matrix Multiplication Distributes Across Addition 105
Suppose A is an m × n matrix and B and C are n × p matrices and D is a p × s matrix. Then
1. A(B + C) = AB + AC
2. (B + C)D = BD + CD
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Theorem MMSMM Matrix Multiplication and Scalar Matrix Multiplication 106
Suppose A is an m × n matrix and B is an n × p matrix. Let α be a scalar. Then α(AB) =
(αA)B = A(αB).
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Theorem MMA Matrix Multiplication is Associative 107
Suppose A is an m × n matrix, B is an n × p matrix and D is a p × s matrix. Then A(BD) =
(AB)D.
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Theorem MMIP Matrix Multiplication and Inner Products 108
If we consider the vectors u, v ∈ Cm as m × 1 matrices then
hu, vi = ut v = u∗ v
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Theorem MMCC Matrix Multiplication and Complex Conjugation 109
Suppose A is an m × n matrix and B is an n × p matrix. Then AB = A B.
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Theorem MMT Matrix Multiplication and Transposes 110
Suppose A is an m × n matrix and B is an n × p matrix. Then (AB)t = B t At .
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Theorem MMAD Matrix Multiplication and Adjoints 111
Suppose A is an m × n matrix and B is an n × p matrix. Then (AB)∗ = B ∗ A∗ .
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Theorem AIP Adjoint and Inner Product 112
Suppose that A is an m × n matrix and x ∈ Cn , y ∈ Cm . Then hAx, yi = hx, A∗ yi.
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Definition HM Hermitian Matrix 113
The square matrix A is Hermitian (or self-adjoint) if A = A∗ .
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Theorem HMIP Hermitian Matrices and Inner Products 114
Suppose that A is a square matrix of size n. Then A is Hermitian if and only if hAx, yi = hx, Ayi
for all x, y ∈ Cn .
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Definition MI Matrix Inverse 115
Suppose A and B are square matrices of size n such that AB = In and BA = In . Then A is
invertible and B is the inverse of A. In this situation, we write B = A−1 .
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Theorem TTMI Two-by-Two Matrix Inverse 116
Suppose
a b
A=
c d
Then A is invertible if and only if ad − bc 6= 0. When A is invertible, then
−1 1 d −b
A =
ad − bc −c a
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Theorem CINM Computing the Inverse of a Nonsingular Matrix 117
Suppose A is a nonsingular square matrix of size n. Create the n × 2n matrix M by placing the
n × n identity matrix In to the right of the matrix A. Let N be a matrix that is row-equivalent
to M and in reduced row-echelon form. Finally, let J be the matrix formed from the final n
columns of N . Then AJ = In .
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Theorem MIU Matrix Inverse is Unique 118
Suppose the square matrix A has an inverse. Then A−1 is unique.
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Theorem SS Socks and Shoes 119
Suppose A and B are invertible matrices of size n. Then AB is an invertible matrix and (AB)−1 =
B −1 A−1 .
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Theorem MIMI Matrix Inverse of a Matrix Inverse 120
Suppose A is an invertible matrix. Then A−1 is invertible and (A−1 )−1 = A.
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Theorem MIT Matrix Inverse of a Transpose 121
Suppose A is an invertible matrix. Then At is invertible and (At )−1 = (A−1 )t .
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Theorem MISM Matrix Inverse of a Scalar Multiple 122
−1 1 −1
Suppose A is an invertible matrix and α is a nonzero scalar. Then (αA) = αA and αA is
invertible.
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Theorem NPNT Nonsingular Product has Nonsingular Terms 123
Suppose that A and B are square matrices of size n. The product AB is nonsingular if and only
if A and B are both nonsingular.
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Theorem OSIS One-Sided Inverse is Sufficient 124
Suppose A and B are square matrices of size n such that AB = In . Then BA = In .
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Theorem NI Nonsingularity is Invertibility 125
Suppose that A is a square matrix. Then A is nonsingular if and only if A is invertible.
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Theorem NME3 Nonsingular Matrix Equivalences, Round 3 126
Suppose that A is a square matrix of size n. The following are equivalent.
1. A is nonsingular.
2. A row-reduces to the identity matrix.
3. The null space of A contains only the zero vector, N (A) = {0}.
4. The linear system LS(A, b) has a unique solution for every possible choice of b.
5. The columns of A are a linearly independent set.
6. A is invertible.
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Theorem SNCM Solution with Nonsingular Coefficient Matrix 127
Suppose that A is nonsingular. Then the unique solution to LS(A, b) is A−1 b.
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Definition UM Unitary Matrices 128
Suppose that U is a square matrix of size n such that U ∗ U = In . Then we say U is unitary.
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Theorem UMI Unitary Matrices are Invertible 129
Suppose that U is a unitary matrix of size n. Then U is nonsingular, and U −1 = U ∗ .
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Theorem CUMOS Columns of Unitary Matrices are Orthonormal Sets 130
Suppose that S = {A1 , A2 , A3 , . . . , An } is the set of columns of a square matrix A of size n.
Then A is a unitary matrix if and only if S is an orthonormal set.
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Theorem UMPIP Unitary Matrices Preserve Inner Products 131
Suppose that U is a unitary matrix of size n and u and v are two vectors from Cn . Then
hU u, U vi = hu, vi and kU vk = kvk
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Definition CSM Column Space of a Matrix 132
Suppose that A is an m × n matrix with columns A1 , A2 , A3 , . . . , An . Then the column space
of A, written C(A), is the subset of Cm containing all linear combinations of the columns of A,
C(A) = h{A1 , A2 , A3 , . . . , An }i
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Theorem CSCS Column Spaces and Consistent Systems 133
Suppose A is an m × n matrix and b is a vector of size m. Then b ∈ C(A) if and only if LS(A, b)
is consistent.
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Theorem BCS Basis of the Column Space 134
Suppose that A is an m×n matrix with columns A1 , A2 , A3 , . . . , An , and B is a row-equivalent
matrix in reduced row-echelon form with r pivot columns. Let D = {d1 , d2 , d3 , . . . , dr } be the
set of indices for the pivot columns of B Let T = {Ad1 , Ad2 , Ad3 , . . . , Adr }. Then
1. T is a linearly independent set.
2. C(A) = hT i.
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Theorem CSNM Column Space of a Nonsingular Matrix 135
Suppose A is a square matrix of size n. Then A is nonsingular if and only if C(A) = Cn .
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Theorem NME4 Nonsingular Matrix Equivalences, Round 4 136
Suppose that A is a square matrix of size n. The following are equivalent.
1. A is nonsingular.
2. A row-reduces to the identity matrix.
3. The null space of A contains only the zero vector, N (A) = {0}.
4. The linear system LS(A, b) has a unique solution for every possible choice of b.
5. The columns of A are a linearly independent set.
6. A is invertible.
7. The column space of A is Cn , C(A) = Cn .
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Definition RSM Row Space of a Matrix 137
Suppose A is an m × n matrix. Then the row space of A, R(A), is the column space of At , i.e.
R(A) = C(At ).
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Theorem REMRS Row-Equivalent Matrices have equal Row Spaces 138
Suppose A and B are row-equivalent matrices. Then R(A) = R(B).
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Theorem BRS Basis for the Row Space 139
Suppose that A is a matrix and B is a row-equivalent matrix in reduced row-echelon form. Let
S be the set of nonzero columns of B t . Then
1. R(A) = hSi.
2. S is a linearly independent set.
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Theorem CSRST Column Space, Row Space, Transpose 140
Suppose A is a matrix. Then C(A) = R(At ).
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Definition LNS Left Null Space 141
Suppose A is an m × n matrix. Then the left null space is defined as L(A) = N (At ) ⊆ Cm .
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Definition EEF Extended Echelon Form 142
Suppose A is an m × n matrix. Extend A on its right side with the addition of an m × m
identity matrix to form an m × (n + m) matrix M . Use row operations to bring M to reduced
row-echelon form and call the result N . N is the extended reduced row-echelon form of A, and
we will standardize on names for five submatrices (B, C, J, K, L) of N .
Let B denote the m × n matrix formed from the first n columns of N and let J denote the m × m
matrix formed from the last m columns of N . Suppose that B has r nonzero rows. Further
partition N by letting C denote the r × n matrix formed from all of the nonzero rows of B. Let
K be the r × m matrix formed from the first r rows of J, while L will be the (m − r) × m matrix
formed from the bottom m − r rows of J. Pictorially,
RREF C K
M = [A|Im ] −−−−→ N = [B|J] =
0 L
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Theorem PEEF Properties of Extended Echelon Form 143
Suppose that A is an m × n matrix and that N is its extended echelon form. Then
1. J is nonsingular.
2. B = JA.
3. If x ∈ Cn and y ∈ Cm , then Ax = y if and only if Bx = Jy.
4. C is in reduced row-echelon form, has no zero rows and has r pivot columns.
5. L is in reduced row-echelon form, has no zero rows and has m − r pivot columns.
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Theorem FS Four Subsets 144
Suppose A is an m × n matrix with extended echelon form N . Suppose the reduced row-echelon
form of A has r nonzero rows. Then C is the submatrix of N formed from the first r rows and
the first n columns and L is the submatrix of N formed from the last m columns and the last
m − r rows. Then
1. The null space of A is the null space of C, N (A) = N (C).
2. The row space of A is the row space of C, R(A) = R(C).
3. The column space of A is the null space of L, C(A) = N (L).
4. The left null space of A is the row space of L, L(A) = R(L).
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Definition VS Vector Space 145
Suppose that V is a set upon which we have defined two operations: (1) vector addition, which
combines two elements of V and is denoted by “+”, and (2) scalar multiplication, which combines
a complex number with an element of V and is denoted by juxtaposition. Then V , along with
the two operations, is a vector space over C if the following ten properties hold.
• AC Additive Closure: If u, v ∈ V , then u + v ∈ V .
• SC Scalar Closure: If α ∈ C and u ∈ V , then αu ∈ V .
• C Commutativity: If u, v ∈ V , then u + v = v + u.
• AA Additive Associativity: If u, v, w ∈ V , then u + (v + w) = (u + v) + w.
• Z Zero Vector: There is a vector, 0, called the zero vector, such that u + 0 = u for all
u∈V.
• AI Additive Inverses: If u ∈ V , then there exists a vector −u ∈ V so that u + (−u) = 0.
• SMA Scalar Multiplication Associativity: If α, β ∈ C and u ∈ V , then α(βu) = (αβ)u.
• DVA Distributivity across Vector Addition: If α ∈ C and u, v ∈ V , then α(u + v) =
αu + αv.
• DSA Distributivity across Scalar Addition: If α, β ∈ C and u ∈ V , then (α+β)u = αu+βu.
• O One: If u ∈ V , then 1u = u.
The objects in V are called vectors, no matter what else they might really be, simply by virtue
of being elements of a vector space.
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Theorem ZVU Zero Vector is Unique 146
Suppose that V is a vector space. The zero vector, 0, is unique.
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Theorem AIU Additive Inverses are Unique 147
Suppose that V is a vector space. For each u ∈ V , the additive inverse, −u, is unique.
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Theorem ZSSM Zero Scalar in Scalar Multiplication 148
Suppose that V is a vector space and u ∈ V . Then 0u = 0.
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Theorem ZVSM Zero Vector in Scalar Multiplication 149
Suppose that V is a vector space and α ∈ C. Then α0 = 0.
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Theorem AISM Additive Inverses from Scalar Multiplication 150
Suppose that V is a vector space and u ∈ V . Then −u = (−1)u.
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Theorem SMEZV Scalar Multiplication Equals the Zero Vector 151
Suppose that V is a vector space and α ∈ C. If αu = 0, then either α = 0 or u = 0.
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Definition S Subspace 152
Suppose that V and W are two vector spaces that have identical definitions of vector addition
and scalar multiplication, and that W is a subset of V , W ⊆ V . Then W is a subspace of V .
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Theorem TSS Testing Subsets for Subspaces 153
Suppose that V is a vector space and W is a subset of V , W ⊆ V . Endow W with the same
operations as V . Then W is a subspace if and only if three conditions are met
1. W is nonempty, W 6= ∅.
2. If x ∈ W and y ∈ W , then x + y ∈ W .
3. If α ∈ C and x ∈ W , then αx ∈ W .
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Definition TS Trivial Subspaces 154
Given the vector space V , the subspaces V and {0} are each called a trivial subspace.
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Theorem NSMS Null Space of a Matrix is a Subspace 155
Suppose that A is an m × n matrix. Then the null space of A, N (A), is a subspace of Cn .
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Definition LC Linear Combination 156
Suppose that V is a vector space. Given n vectors u1 , u2 , u3 , . . . , un and n scalars
α1 , α2 , α3 , . . . , αn , their linear combination is the vector
α1 u1 + α2 u2 + α3 u3 + · · · + αn un .
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Definition SS Span of a Set 157
Suppose that V is a vector space. Given a set of vectors S = {u1 , u2 , u3 , . . . , ut }, their span,
hSi, is the set of all possible linear combinations of u1 , u2 , u3 , . . . , ut . Symbolically,
hSi = { α1 u1 + α2 u2 + α3 u3 + · · · + αt ut | αi ∈ C, 1 ≤ i ≤ t}
( t )
X
= αi ui αi ∈ C, 1 ≤ i ≤ t
i=1
c 2004—2015 Robert A. Beezer, GFDL License
Theorem SSS Span of a Set is a Subspace 158
Suppose V is a vector space. Given a set of vectors S = {u1 , u2 , u3 , . . . , ut } ⊆ V , their span,
hSi, is a subspace.
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Theorem CSMS Column Space of a Matrix is a Subspace 159
Suppose that A is an m × n matrix. Then C(A) is a subspace of Cm .
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Theorem RSMS Row Space of a Matrix is a Subspace 160
Suppose that A is an m × n matrix. Then R(A) is a subspace of Cn .
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Theorem LNSMS Left Null Space of a Matrix is a Subspace 161
Suppose that A is an m × n matrix. Then L(A) is a subspace of Cm .
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Definition RLD Relation of Linear Dependence 162
Suppose that V is a vector space. Given a set of vectors S = {u1 , u2 , u3 , . . . , un }, an equation
of the form
α1 u1 + α2 u2 + α3 u3 + · · · + αn un = 0
is a relation of linear dependence on S. If this equation is formed in a trivial fashion, i.e. αi = 0,
1 ≤ i ≤ n, then we say it is a trivial relation of linear dependence on S.
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Definition LI Linear Independence 163
Suppose that V is a vector space. The set of vectors S = {u1 , u2 , u3 , . . . , un } from V is linearly
dependent if there is a relation of linear dependence on S that is not trivial. In the case where
the only relation of linear dependence on S is the trivial one, then S is a linearly independent
set of vectors.
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Definition SSVS Spanning Set of a Vector Space 164
Suppose V is a vector space. A subset S of V is a spanning set of V if hSi = V . In this case, we
also frequently say S spans V .
c 2004—2015 Robert A. Beezer, GFDL License
Theorem VRRB Vector Representation Relative to a Basis 165
Suppose that V is a vector space and B = {v1 , v2 , v3 , . . . , vm } is a linearly independent set
that spans V . Let w be any vector in V . Then there exist unique scalars a1 , a2 , a3 , . . . , am such
that
w = a1 v1 + a2 v2 + a3 v3 + · · · + am vm .
c 2004—2015 Robert A. Beezer, GFDL License
Definition B Basis 166
Suppose V is a vector space. Then a subset S ⊆ V is a basis of V if it is linearly independent
and spans V .
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Theorem SUVB Standard Unit Vectors are a Basis 167
The set of standard unit vectors for Cm (Definition SUV), B = { ei | 1 ≤ i ≤ m} is a basis for the
vector space Cm .
c 2004—2015 Robert A. Beezer, GFDL License
Theorem CNMB Columns of Nonsingular Matrix are a Basis 168
Suppose that A is a square matrix of size m. Then the columns of A are a basis of Cm if and
only if A is nonsingular.
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Theorem NME5 Nonsingular Matrix Equivalences, Round 5 169
Suppose that A is a square matrix of size n. The following are equivalent.
1. A is nonsingular.
2. A row-reduces to the identity matrix.
3. The null space of A contains only the zero vector, N (A) = {0}.
4. The linear system LS(A, b) has a unique solution for every possible choice of b.
5. The columns of A are a linearly independent set.
6. A is invertible.
7. The column space of A is Cn , C(A) = Cn .
8. The columns of A are a basis for Cn .
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Theorem COB Coordinates and Orthonormal Bases 170
Suppose that B = {v1 , v2 , v3 , . . . , vp } is an orthonormal basis of the subspace W of Cm . For
any w ∈ W ,
w = hv1 , wi v1 + hv2 , wi v2 + hv3 , wi v3 + · · · + hvp , wi vp
c 2004—2015 Robert A. Beezer, GFDL License
Theorem UMCOB Unitary Matrices Convert Orthonormal Bases 171
Let A be an n × n matrix and B = {x1 , x2 , x3 , . . . , xn } be an orthonormal basis of Cn . Define
C = {Ax1 , Ax2 , Ax3 , . . . , Axn }
Then A is a unitary matrix if and only if C is an orthonormal basis of Cn .
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Definition D Dimension 172
Suppose that V is a vector space and {v1 , v2 , v3 , . . . , vt } is a basis of V . Then the dimension
of V is defined by dim (V ) = t. If V has no finite bases, we say V has infinite dimension.
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Theorem SSLD Spanning Sets and Linear Dependence 173
Suppose that S = {v1 , v2 , v3 , . . . , vt } is a finite set of vectors which spans the vector space V .
Then any set of t + 1 or more vectors from V is linearly dependent.
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Theorem BIS Bases have Identical Sizes 174
Suppose that V is a vector space with a finite basis B and a second basis C. Then B and C have
the same size.
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Theorem DCM Dimension of Cm 175
The dimension of Cm (Example VSCV) is m.
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Theorem DP Dimension of Pn 176
The dimension of Pn (Example VSP) is n + 1.
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Theorem DM Dimension of Mmn 177
The dimension of Mmn (Example VSM) is mn.
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Definition NOM Nullity Of a Matrix 178
Suppose that A is an m × n matrix. Then the nullity of A is the dimension of the null space of
A, n (A) = dim (N (A)).
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Definition ROM Rank Of a Matrix 179
Suppose that A is an m × n matrix. Then the rank of A is the dimension of the column space of
A, r (A) = dim (C(A)).
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Theorem CRN Computing Rank and Nullity 180
Suppose that A is an m×n matrix and B is a row-equivalent matrix in reduced row-echelon form.
Let r denote the number of pivot columns (or the number of nonzero rows). Then r (A) = r and
n (A) = n − r.
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Theorem RPNC Rank Plus Nullity is Columns 181
Suppose that A is an m × n matrix. Then r (A) + n (A) = n.
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Theorem RNNM Rank and Nullity of a Nonsingular Matrix 182
Suppose that A is a square matrix of size n. The following are equivalent.
1. A is nonsingular.
2. The rank of A is n, r (A) = n.
3. The nullity of A is zero, n (A) = 0.
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Theorem NME6 Nonsingular Matrix Equivalences, Round 6 183
Suppose that A is a square matrix of size n. The following are equivalent.
1. A is nonsingular.
2. A row-reduces to the identity matrix.
3. The null space of A contains only the zero vector, N (A) = {0}.
4. The linear system LS(A, b) has a unique solution for every possible choice of b.
5. The columns of A are a linearly independent set.
6. A is invertible.
7. The column space of A is Cn , C(A) = Cn .
8. The columns of A are a basis for Cn .
9. The rank of A is n, r (A) = n.
10. The nullity of A is zero, n (A) = 0.
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Theorem ELIS Extending Linearly Independent Sets 184
Suppose V is a vector space and S is a linearly independent set of vectors from V . Suppose w
is a vector such that w 6∈ hSi. Then the set S 0 = S ∪ {w} is linearly independent.
c 2004—2015 Robert A. Beezer, GFDL License
Theorem G Goldilocks 185
Suppose that V is a vector space of dimension t. Let S = {v1 , v2 , v3 , . . . , vm } be a set of
vectors from V . Then
1. If m > t, then S is linearly dependent.
2. If m < t, then S does not span V .
3. If m = t and S is linearly independent, then S spans V .
4. If m = t and S spans V , then S is linearly independent.
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Theorem PSSD Proper Subspaces have Smaller Dimension 186
Suppose that U and V are subspaces of the vector space W , such that U ( V . Then dim (U ) <
dim (V ).
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Theorem EDYES Equal Dimensions Yields Equal Subspaces 187
Suppose that U and V are subspaces of the vector space W , such that U ⊆ V and dim (U ) =
dim (V ). Then U = V .
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Theorem RMRT Rank of a Matrix is the Rank of the Transpose 188
Suppose A is an m × n matrix. Then r (A) = r (At ).
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Theorem DFS Dimensions of Four Subspaces 189
Suppose that A is an m × n matrix, and B is a row-equivalent matrix in reduced row-echelon
form with r nonzero rows. Then
1. dim (N (A)) = n − r
2. dim (C(A)) = r
3. dim (R(A)) = r
4. dim (L(A)) = m − r
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Definition ELEM Elementary Matrices 190
1. For i 6= j, Ei,j is the square matrix of size n with
0
k 6= i, k 6= j, ` 6= k
1 k 6= i, k 6= j, ` = k
0 k = i, ` 6= j
[Ei,j ]k` =
1 k = i, ` = j
0 k = j, ` 6= i
1 k = j, ` = i
2. For α 6= 0, Ei (α) is the square matrix of size n with
0
` 6= k
[Ei (α)]k` = 1 k 6= i, ` = k
α k = i, ` = i
3. For i 6= j, Ei,j (α) is the square matrix of size n with
0 k 6 j, ` 6= k
=
1 k 6= j, ` = k
[Ei,j (α)]k` = 0 k = j, ` 6= i, ` 6= j
1
k = j, ` = j
α k = j, ` = i
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Theorem EMDRO Elementary Matrices Do Row Operations 191
Suppose that A is an m × n matrix, and B is a matrix of the same size that is obtained from
A by a single row operation (Definition RO). Then there is an elementary matrix of size m that
will convert A to B via matrix multiplication on the left. More precisely,
1. If the row operation swaps rows i and j, then B = Ei,j A.
2. If the row operation multiplies row i by α, then B = Ei (α) A.
3. If the row operation multiplies row i by α and adds the result to row j, then B = Ei,j (α) A.
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Theorem EMN Elementary Matrices are Nonsingular 192
If E is an elementary matrix, then E is nonsingular.
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Theorem NMPEM Nonsingular Matrices are Products of Elementary Matrices193
Suppose that A is a nonsingular matrix. Then there exists elementary matrices
E1 , E2 , E3 , . . . , Et so that A = E1 E2 E3 . . . Et .
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Definition SM SubMatrix 194
Suppose that A is an m × n matrix. Then the submatrix A (i|j) is the (m − 1) × (n − 1) matrix
obtained from A by removing row i and column j.
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Definition DM Determinant of a Matrix 195
Suppose A is a square matrix. Then its determinant, det (A) = |A|, is an element of C defined
recursively by:
1. If A is a 1 × 1 matrix, then det (A) = [A]11 .
2. If A is a matrix of size n with n ≥ 2, then
det (A) = [A]11 det (A (1|1)) − [A]12 det (A (1|2)) + [A]13 det (A (1|3)) −
[A]14 det (A (1|4)) + · · · + (−1)n+1 [A]1n det (A (1|n))
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Theorem DMST Determinant
of Matrices of Size Two 196
a b
Suppose that A = . Then det (A) = ad − bc.
c d
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Theorem DER Determinant Expansion about Rows 197
Suppose that A is a square matrix of size n. Then for 1 ≤ i ≤ n
det (A) = (−1)i+1 [A]i1 det (A (i|1)) + (−1)i+2 [A]i2 det (A (i|2))
+ (−1)i+3 [A]i3 det (A (i|3)) + · · · + (−1)i+n [A]in det (A (i|n))
which is known as expansion about row i.
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Theorem DT Determinant of the Transpose 198
Suppose that A is a square matrix. Then det (At ) = det (A).
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Theorem DEC Determinant Expansion about Columns 199
Suppose that A is a square matrix of size n. Then for 1 ≤ j ≤ n
det (A) = (−1)1+j [A]1j det (A (1|j)) + (−1)2+j [A]2j det (A (2|j))
+ (−1)3+j [A]3j det (A (3|j)) + · · · + (−1)n+j [A]nj det (A (n|j))
which is known as expansion about column j.
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Theorem DZRC Determinant with Zero Row or Column 200
Suppose that A is a square matrix with a row where every entry is zero, or a column where every
entry is zero. Then det (A) = 0.
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Theorem DRCS Determinant for Row or Column Swap 201
Suppose that A is a square matrix. Let B be the square matrix obtained from A by interchanging
the location of two rows, or interchanging the location of two columns. Then det (B) = − det (A).
c 2004—2015 Robert A. Beezer, GFDL License
Theorem DRCM Determinant for Row or Column Multiples 202
Suppose that A is a square matrix. Let B be the square matrix obtained from A by multiplying
a single row by the scalar α, or by multiplying a single column by the scalar α. Then det (B) =
α det (A).
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Theorem DERC Determinant with Equal Rows or Columns 203
Suppose that A is a square matrix with two equal rows, or two equal columns. Then det (A) = 0.
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Theorem DRCMA Determinant for Row or Column Multiples and Addition 204
Suppose that A is a square matrix. Let B be the square matrix obtained from A by multiplying
a row by the scalar α and then adding it to another row, or by multiplying a column by the
scalar α and then adding it to another column. Then det (B) = det (A).
c 2004—2015 Robert A. Beezer, GFDL License
Theorem DIM Determinant of the Identity Matrix 205
For every n ≥ 1, det (In ) = 1.
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Theorem DEM Determinants of Elementary Matrices 206
For the three possible versions of an elementary matrix (Definition ELEM) we have the determi-
nants,
1. det (Ei,j ) = −1
2. det (Ei (α)) = α
3. det (Ei,j (α)) = 1
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Theorem DEMMM Determinants, Elementary Matrices, Matrix Multiplication207
Suppose that A is a square matrix of size n and E is any elementary matrix of size n. Then
det (EA) = det (E) det (A)
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Theorem SMZD Singular Matrices have Zero Determinants 208
Let A be a square matrix. Then A is singular if and only if det (A) = 0.
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Theorem NME7 Nonsingular Matrix Equivalences, Round 7 209
Suppose that A is a square matrix of size n. The following are equivalent.
1. A is nonsingular.
2. A row-reduces to the identity matrix.
3. The null space of A contains only the zero vector, N (A) = {0}.
4. The linear system LS(A, b) has a unique solution for every possible choice of b.
5. The columns of A are a linearly independent set.
6. A is invertible.
7. The column space of A is Cn , C(A) = Cn .
8. The columns of A are a basis for Cn .
9. The rank of A is n, r (A) = n.
10. The nullity of A is zero, n (A) = 0.
11. The determinant of A is nonzero, det (A) 6= 0.
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Theorem DRMM Determinant Respects Matrix Multiplication 210
Suppose that A and B are square matrices of the same size. Then det (AB) = det (A) det (B).
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Definition EEM Eigenvalues and Eigenvectors of a Matrix 211
Suppose that A is a square matrix of size n, x 6= 0 is a vector in Cn , and λ is a scalar in C. Then
we say x is an eigenvector of A with eigenvalue λ if
Ax = λx
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Theorem EMHE Every Matrix Has an Eigenvalue 212
Suppose A is a square matrix. Then A has at least one eigenvalue.
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Definition CP Characteristic Polynomial 213
Suppose that A is a square matrix of size n. Then the characteristic polynomial of A is the
polynomial pA (x) defined by
pA (x) = det (A − xIn )
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Theorem EMRCP Eigenvalues of a Matrix are Roots of Characteristic Polynomials
214
Suppose A is a square matrix. Then λ is an eigenvalue of A if and only if pA (λ) = 0.
c 2004—2015 Robert A. Beezer, GFDL License
Definition EM Eigenspace of a Matrix 215
Suppose that A is a square matrix and λ is an eigenvalue of A. Then the eigenspace of A for λ,
EA (λ), is the set of all the eigenvectors of A for λ, together with the inclusion of the zero vector.
c 2004—2015 Robert A. Beezer, GFDL License
Theorem EMS Eigenspace for a Matrix is a Subspace 216
Suppose A is a square matrix of size n and λ is an eigenvalue of A. Then the eigenspace EA (λ)
is a subspace of the vector space Cn .
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Theorem EMNS Eigenspace of a Matrix is a Null Space 217
Suppose A is a square matrix of size n and λ is an eigenvalue of A. Then
EA (λ) = N (A − λIn )
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Definition AME Algebraic Multiplicity of an Eigenvalue 218
Suppose that A is a square matrix and λ is an eigenvalue of A. Then the algebraic multiplicity
of λ, αA (λ), is the highest power of (x − λ) that divides the characteristic polynomial, pA (x).
c 2004—2015 Robert A. Beezer, GFDL License
Definition GME Geometric Multiplicity of an Eigenvalue 219
Suppose that A is a square matrix and λ is an eigenvalue of A. Then the geometric multiplicity
of λ, γA (λ), is the dimension of the eigenspace EA (λ).
c 2004—2015 Robert A. Beezer, GFDL License
Theorem EDELI Eigenvectors with Distinct Eigenvalues are Linearly Independent
220
Suppose that A is an n × n square matrix and S = {x1 , x2 , x3 , . . . , xp } is a set of eigenvectors
with eigenvalues λ1 , λ2 , λ3 , . . . , λp such that λi 6= λj whenever i 6= j. Then S is a linearly
independent set.
c 2004—2015 Robert A. Beezer, GFDL License
Theorem SMZE Singular Matrices have Zero Eigenvalues 221
Suppose A is a square matrix. Then A is singular if and only if λ = 0 is an eigenvalue of A.
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Theorem NME8 Nonsingular Matrix Equivalences, Round 8 222
Suppose that A is a square matrix of size n. The following are equivalent.
1. A is nonsingular.
2. A row-reduces to the identity matrix.
3. The null space of A contains only the zero vector, N (A) = {0}.
4. The linear system LS(A, b) has a unique solution for every possible choice of b.
5. The columns of A are a linearly independent set.
6. A is invertible.
7. The column space of A is Cn , C(A) = Cn .
8. The columns of A are a basis for Cn .
9. The rank of A is n, r (A) = n.
10. The nullity of A is zero, n (A) = 0.
11. The determinant of A is nonzero, det (A) 6= 0.
12. λ = 0 is not an eigenvalue of A.
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Theorem ESMM Eigenvalues of a Scalar Multiple of a Matrix 223
Suppose A is a square matrix and λ is an eigenvalue of A. Then αλ is an eigenvalue of αA.
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Theorem EOMP Eigenvalues Of Matrix Powers 224
Suppose A is a square matrix, λ is an eigenvalue of A, and s ≥ 0 is an integer. Then λs is an
eigenvalue of As .
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Theorem EPM Eigenvalues of the Polynomial of a Matrix 225
Suppose A is a square matrix and λ is an eigenvalue of A. Let q(x) be a polynomial in the
variable x. Then q(λ) is an eigenvalue of the matrix q(A).
c 2004—2015 Robert A. Beezer, GFDL License
Theorem EIM Eigenvalues of the Inverse of a Matrix 226
Suppose A is a square nonsingular matrix and λ is an eigenvalue of A. Then λ−1 is an eigenvalue
of the matrix A−1 .
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Theorem ETM Eigenvalues of the Transpose of a Matrix 227
Suppose A is a square matrix and λ is an eigenvalue of A. Then λ is an eigenvalue of the matrix
At .
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Theorem ERMCP Eigenvalues of Real Matrices come in Conjugate Pairs 228
Suppose A is a square matrix with real entries and x is an eigenvector of A for the eigenvalue λ.
Then x is an eigenvector of A for the eigenvalue λ.
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Theorem DCP Degree of the Characteristic Polynomial 229
Suppose that A is a square matrix of size n. Then the characteristic polynomial of A, pA (x),
has degree n.
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Theorem NEM Number of Eigenvalues of a Matrix 230
Suppose that λ1 , λ2 , λ3 , . . . , λk are the distinct eigenvalues of a square matrix A of size n. Then
k
X
αA (λi ) = n
i=1
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Theorem ME Multiplicities of an Eigenvalue 231
Suppose that A is a square matrix of size n and λ is an eigenvalue. Then
1 ≤ γA (λ) ≤ αA (λ) ≤ n
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Theorem MNEM Maximum Number of Eigenvalues of a Matrix 232
Suppose that A is a square matrix of size n. Then A cannot have more than n distinct eigenvalues.
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Theorem HMRE Hermitian Matrices have Real Eigenvalues 233
Suppose that A is a Hermitian matrix and λ is an eigenvalue of A. Then λ ∈ R.
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Theorem HMOE Hermitian Matrices have Orthogonal Eigenvectors 234
Suppose that A is a Hermitian matrix and x and y are two eigenvectors of A for different
eigenvalues. Then x and y are orthogonal vectors.
c 2004—2015 Robert A. Beezer, GFDL License
Definition SIM Similar Matrices 235
Suppose A and B are two square matrices of size n. Then A and B are similar if there exists a
nonsingular matrix of size n, S, such that A = S −1 BS.
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Theorem SER Similarity is an Equivalence Relation 236
Suppose A, B and C are square matrices of size n. Then
1. A is similar to A. (Reflexive)
2. If A is similar to B, then B is similar to A. (Symmetric)
3. If A is similar to B and B is similar to C, then A is similar to C. (Transitive)
c 2004—2015 Robert A. Beezer, GFDL License
Theorem SMEE Similar Matrices have Equal Eigenvalues 237
Suppose A and B are similar matrices. Then the characteristic polynomials of A and B are
equal, that is, pA (x) = pB (x).
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Definition DIM Diagonal Matrix 238
Suppose that A is a square matrix. Then A is a diagonal matrix if [A]ij = 0 whenever i 6= j.
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Definition DZM Diagonalizable Matrix 239
Suppose A is a square matrix. Then A is diagonalizable if A is similar to a diagonal matrix.
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Theorem DC Diagonalization Characterization 240
Suppose A is a square matrix of size n. Then A is diagonalizable if and only if there exists a
linearly independent set S that contains n eigenvectors of A.
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Theorem DMFE Diagonalizable Matrices have Full Eigenspaces 241
Suppose A is a square matrix. Then A is diagonalizable if and only if γA (λ) = αA (λ) for every
eigenvalue λ of A.
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Theorem DED Distinct Eigenvalues implies Diagonalizable 242
Suppose A is a square matrix of size n with n distinct eigenvalues. Then A is diagonalizable.
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Definition LT Linear Transformation 243
A linear transformation, T : U → V , is a function that carries elements of the vector space U
(called the domain) to the vector space V (called the codomain), and which has two additional
properties
1. T (u1 + u2 ) = T (u1 ) + T (u2 ) for all u1 , u2 ∈ U
2. T (αu) = αT (u) for all u ∈ U and all α ∈ C
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Theorem LTTZZ Linear Transformations Take Zero to Zero 244
Suppose T : U → V is a linear transformation. Then T (0) = 0.
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Theorem MBLT Matrices Build Linear Transformations 245
Suppose that A is an m × n matrix. Define a function T : Cn → Cm by T (x) = Ax. Then T is
a linear transformation.
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Theorem MLTCV Matrix of a Linear Transformation, Column Vectors 246
Suppose that T : Cn → Cm is a linear transformation. Then there is an m × n matrix A such
that T (x) = Ax.
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Theorem LTLC Linear Transformations and Linear Combinations 247
Suppose that T : U → V is a linear transformation, u1 , u2 , u3 , . . . , ut are vectors from U and
a1 , a2 , a3 , . . . , at are scalars from C. Then
T (a1 u1 + a2 u2 + a3 u3 + · · · + at ut ) = a1 T (u1 ) + a2 T (u2 ) + a3 T (u3 ) + · · · + at T (ut )
c 2004—2015 Robert A. Beezer, GFDL License
Theorem LTDB Linear Transformation Defined on a Basis 248
Suppose U is a vector space with basis B = {u1 , u2 , u3 , . . . , un } and the vector space V con-
tains the vectors v1 , v2 , v3 , . . . , vn (which may not be distinct). Then there is a unique linear
transformation, T : U → V , such that T (ui ) = vi , 1 ≤ i ≤ n.
c 2004—2015 Robert A. Beezer, GFDL License
Definition PI Pre-Image 249
Suppose that T : U → V is a linear transformation. For each v, define the pre-image of v to be
the subset of U given by
T −1 (v) = { u ∈ U | T (u) = v}
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Definition LTA Linear Transformation Addition 250
Suppose that T : U → V and S : U → V are two linear transformations with the same domain
and codomain. Then their sum is the function T + S : U → V whose outputs are defined by
(T + S) (u) = T (u) + S (u)
c 2004—2015 Robert A. Beezer, GFDL License
Theorem SLTLT Sum of Linear Transformations is a Linear Transformation 251
Suppose that T : U → V and S : U → V are two linear transformations with the same domain
and codomain. Then T + S : U → V is a linear transformation.
c 2004—2015 Robert A. Beezer, GFDL License
Definition LTSM Linear Transformation Scalar Multiplication 252
Suppose that T : U → V is a linear transformation and α ∈ C. Then the scalar multiple is the
function αT : U → V whose outputs are defined by
(αT ) (u) = αT (u)
c 2004—2015 Robert A. Beezer, GFDL License
Theorem MLTLT Multiple of a Linear Transformation is a Linear Transformation
253
Suppose that T : U → V is a linear transformation and α ∈ C. Then (αT ) : U → V is a linear
transformation.
c 2004—2015 Robert A. Beezer, GFDL License
Theorem VSLT Vector Space of Linear Transformations 254
Suppose that U and V are vector spaces. Then the set of all linear transformations from U
to V , LT (U, V ), is a vector space when the operations are those given in Definition LTA and
Definition LTSM.
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Definition LTC Linear Transformation Composition 255
Suppose that T : U → V and S : V → W are linear transformations. Then the composition of S
and T is the function (S ◦ T ) : U → W whose outputs are defined by
(S ◦ T ) (u) = S (T (u))
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Theorem CLTLT Composition of Linear Transformations is a Linear Transforma-
tion 256
Suppose that T : U → V and S : V → W are linear transformations. Then (S ◦ T ) : U → W is a
linear transformation.
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Definition ILT Injective Linear Transformation 257
Suppose T : U → V is a linear transformation. Then T is injective if whenever T (x) = T (y),
then x = y.
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Definition KLT Kernel of a Linear Transformation 258
Suppose T : U → V is a linear transformation. Then the kernel of T is the set
K(T ) = { u ∈ U | T (u) = 0}
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Theorem KLTS Kernel of a Linear Transformation is a Subspace 259
Suppose that T : U → V is a linear transformation. Then the kernel of T , K(T ), is a subspace of
U.
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Theorem KPI Kernel and Pre-Image 260
Suppose T : U → V is a linear transformation and v ∈ V . If the preimage T −1 (v) is nonempty,
and u ∈ T −1 (v) then
T −1 (v) = { u + z| z ∈ K(T )} = u + K(T )
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Theorem KILT Kernel of an Injective Linear Transformation 261
Suppose that T : U → V is a linear transformation. Then T is injective if and only if the kernel
of T is trivial, K(T ) = {0}.
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Theorem ILTLI Injective Linear Transformations and Linear Independence 262
Suppose that T : U → V is an injective linear transformation and
S = {u1 , u2 , u3 , . . . , ut }
is a linearly independent subset of U . Then
R = {T (u1 ) , T (u2 ) , T (u3 ) , . . . , T (ut )}
is a linearly independent subset of V .
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Theorem ILTB Injective Linear Transformations and Bases 263
Suppose that T : U → V is a linear transformation and
B = {u1 , u2 , u3 , . . . , um }
is a basis of U . Then T is injective if and only if
C = {T (u1 ) , T (u2 ) , T (u3 ) , . . . , T (um )}
is a linearly independent subset of V .
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Theorem ILTD Injective Linear Transformations and Dimension 264
Suppose that T : U → V is an injective linear transformation. Then dim (U ) ≤ dim (V ).
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Theorem CILTI Composition of Injective Linear Transformations is Injective 265
Suppose that T : U → V and S : V → W are injective linear transformations. Then (S ◦ T ) : U →
W is an injective linear transformation.
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Definition SLT Surjective Linear Transformation 266
Suppose T : U → V is a linear transformation. Then T is surjective if for every v ∈ V there
exists a u ∈ U so that T (u) = v.
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Definition RLT Range of a Linear Transformation 267
Suppose T : U → V is a linear transformation. Then the range of T is the set
R(T ) = { T (u)| u ∈ U }
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Theorem RLTS Range of a Linear Transformation is a Subspace 268
Suppose that T : U → V is a linear transformation. Then the range of T , R(T ), is a subspace of
V.
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Theorem RSLT Range of a Surjective Linear Transformation 269
Suppose that T : U → V is a linear transformation. Then T is surjective if and only if the range
of T equals the codomain, R(T ) = V .
c 2004—2015 Robert A. Beezer, GFDL License
Theorem SSRLT Spanning Set for Range of a Linear Transformation 270
Suppose that T : U → V is a linear transformation and
S = {u1 , u2 , u3 , . . . , ut }
spans U . Then
R = {T (u1 ) , T (u2 ) , T (u3 ) , . . . , T (ut )}
spans R(T ).
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Theorem RPI Range and Pre-Image 271
Suppose that T : U → V is a linear transformation. Then
v ∈ R(T ) if and only if T −1 (v) 6= ∅
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Theorem SLTB Surjective Linear Transformations and Bases 272
Suppose that T : U → V is a linear transformation and
B = {u1 , u2 , u3 , . . . , um }
is a basis of U . Then T is surjective if and only if
C = {T (u1 ) , T (u2 ) , T (u3 ) , . . . , T (um )}
is a spanning set for V .
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Theorem SLTD Surjective Linear Transformations and Dimension 273
Suppose that T : U → V is a surjective linear transformation. Then dim (U ) ≥ dim (V ).
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Theorem CSLTS Composition of Surjective Linear Transformations is Surjective
274
Suppose that T : U → V and S : V → W are surjective linear transformations. Then (S◦T ) : U →
W is a surjective linear transformation.
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Definition IDLT Identity Linear Transformation 275
The identity linear transformation on the vector space W is defined as
IW : W → W, IW (w) = w
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Definition IVLT Invertible Linear Transformations 276
Suppose that T : U → V is a linear transformation. If there is a function S : V → U such that
S ◦ T = IU T ◦ S = IV
then T is invertible. In this case, we call S the inverse of T and write S = T −1 .
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Theorem ILTLT Inverse of a Linear Transformation is a Linear Transformation277
Suppose that T : U → V is an invertible linear transformation. Then the function T −1 : V → U
is a linear transformation.
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Theorem IILT Inverse of an Invertible Linear Transformation 278
Suppose that T : U → V is an invertible linear transformation. Then T −1 is an invertible linear
−1
transformation and T −1 = T.
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Theorem ILTIS Invertible Linear Transformations are Injective and Surjective279
Suppose T : U → V is a linear transformation. Then T is invertible if and only if T is injective
and surjective.
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Theorem CIVLT Composition of Invertible Linear Transformations 280
Suppose that T : U → V and S : V → W are invertible linear transformations. Then the compo-
sition, (S ◦ T ) : U → W is an invertible linear transformation.
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Theorem ICLT Inverse of a Composition of Linear Transformations 281
Suppose that T : U → V and S : V → W are invertible linear transformations. Then S ◦ T is
−1
invertible and (S ◦ T ) = T −1 ◦ S −1 .
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Definition IVS Isomorphic Vector Spaces 282
Two vector spaces U and V are isomorphic if there exists an invertible linear transformation
T with domain U and codomain V , T : U → V . In this case, we write U ∼ = V , and the linear
transformation T is known as an isomorphism between U and V .
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Theorem IVSED Isomorphic Vector Spaces have Equal Dimension 283
Suppose U and V are isomorphic vector spaces. Then dim (U ) = dim (V ).
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Definition ROLT Rank Of a Linear Transformation 284
Suppose that T : U → V is a linear transformation. Then the rank of T , r (T ), is the dimension
of the range of T ,
r (T ) = dim (R(T ))
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Definition NOLT Nullity Of a Linear Transformation 285
Suppose that T : U → V is a linear transformation. Then the nullity of T , n (T ), is the dimension
of the kernel of T ,
n (T ) = dim (K(T ))
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Theorem ROSLT Rank Of a Surjective Linear Transformation 286
Suppose that T : U → V is a linear transformation. Then the rank of T is the dimension of V ,
r (T ) = dim (V ), if and only if T is surjective.
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Theorem NOILT Nullity Of an Injective Linear Transformation 287
Suppose that T : U → V is a linear transformation. Then the nullity of T is zero, n (T ) = 0, if
and only if T is injective.
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Theorem RPNDD Rank Plus Nullity is Domain Dimension 288
Suppose that T : U → V is a linear transformation. Then
r (T ) + n (T ) = dim (U )
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Definition VR Vector Representation 289
Suppose that V is a vector space with a basis B = {v1 , v2 , v3 , . . . , vn }. Define a function
ρB : V → Cn as follows. For w ∈ V define the column vector ρB (w) ∈ Cn by
w = [ρB (w)]1 v1 + [ρB (w)]2 v2 + [ρB (w)]3 v3 + · · · + [ρB (w)]n vn
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Theorem VRLT Vector Representation is a Linear Transformation 290
The function ρB (Definition VR) is a linear transformation.
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Theorem VRI Vector Representation is Injective 291
The function ρB (Definition VR) is an injective linear transformation.
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Theorem VRS Vector Representation is Surjective 292
The function ρB (Definition VR) is a surjective linear transformation.
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Theorem VRILT Vector Representation is an Invertible Linear Transformation293
The function ρB (Definition VR) is an invertible linear transformation.
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Theorem CFDVS Characterization of Finite Dimensional Vector Spaces 294
Suppose that V is a vector space with dimension n. Then V is isomorphic to Cn .
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Theorem IFDVS Isomorphism of Finite Dimensional Vector Spaces 295
Suppose U and V are both finite-dimensional vector spaces. Then U and V are isomorphic if
and only if dim (U ) = dim (V ).
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Theorem CLI Coordinatization and Linear Independence 296
Suppose that U is a vector space with a basis B of size n. Then
S = {u1 , u2 , u3 , . . . , uk }
is a linearly independent subset of U if and only if
R = {ρB (u1 ) , ρB (u2 ) , ρB (u3 ) , . . . , ρB (uk )}
is a linearly independent subset of Cn .
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Theorem CSS Coordinatization and Spanning Sets 297
Suppose that U is a vector space with a basis B of size n. Then
u ∈ h{u1 , u2 , u3 , . . . , uk }i
if and only if
ρB (u) ∈ h{ρB (u1 ) , ρB (u2 ) , ρB (u3 ) , . . . , ρB (uk )}i
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Definition MR Matrix Representation 298
Suppose that T : U → V is a linear transformation, B = {u1 , u2 , u3 , . . . , un } is a basis for U of
size n, and C is a basis for V of size m. Then the matrix representation of T relative to B and
C is the m × n matrix,
T
MB,C = [ ρC (T (u1 ))| ρC (T (u2 ))| ρC (T (u3 ))| . . . |ρC (T (un )) ]
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Theorem FTMR Fundamental Theorem of Matrix Representation 299
Suppose that T : U → V is a linear transformation, B is a basis for U , C is a basis for V and
T
MB,C is the matrix representation of T relative to B and C. Then, for any u ∈ U ,
T
ρC (T (u)) = MB,C (ρB (u))
or equivalently
T (u) = ρ−1 T
C MB,C (ρB (u))
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Theorem MRSLT Matrix Representation of a Sum of Linear Transformations 300
Suppose that T : U → V and S : U → V are linear transformations, B is a basis of U and C is a
basis of V . Then
T +S T S
MB,C = MB,C + MB,C
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Theorem MRMLT Matrix Representation of a Multiple of a Linear Transformation
301
Suppose that T : U → V is a linear transformation, α ∈ C, B is a basis of U and C is a basis of
V . Then
αT T
MB,C = αMB,C
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Theorem MRCLT Matrix Representation of a Composition of Linear Transforma-
tions 302
Suppose that T : U → V and S : V → W are linear transformations, B is a basis of U , C is a
basis of V , and D is a basis of W . Then
S◦T S T
MB,D = MC,D MB,C
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Theorem KNSI Kernel and Null Space Isomorphism 303
Suppose that T : U → V is a linear transformation, B is a basis for U of size n, and C is a basis
T
for V . Then the kernel of T is isomorphic to the null space of MB,C ,
K(T ) ∼ T
= N MB,C
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Theorem RCSI Range and Column Space Isomorphism 304
Suppose that T : U → V is a linear transformation, B is a basis for U of size n, and C is a basis
T
for V of size m. Then the range of T is isomorphic to the column space of MB,C ,
R(T ) ∼ T
= C MB,C
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Theorem IMR Invertible Matrix Representations 305
Suppose that T : U → V is a linear transformation, B is a basis for U and C is a basis for V .
Then T is an invertible linear transformation if and only if the matrix representation of T relative
T
to B and C, MB,C is an invertible matrix. When T is invertible,
T −1
T
−1
MC,B = MB,C
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Theorem IMILT Invertible Matrices, Invertible Linear Transformation 306
Suppose that A is a square matrix of size n and T : Cn → Cn is the linear transformation
defined by T (x) = Ax. Then A is an invertible matrix if and only if T is an invertible linear
transformation.
c 2004—2015 Robert A. Beezer, GFDL License
Theorem NME9 Nonsingular Matrix Equivalences, Round 9 307
Suppose that A is a square matrix of size n. The following are equivalent.
1. A is nonsingular.
2. A row-reduces to the identity matrix.
3. The null space of A contains only the zero vector, N (A) = {0}.
4. The linear system LS(A, b) has a unique solution for every possible choice of b.
5. The columns of A are a linearly independent set.
6. A is invertible.
7. The column space of A is Cn , C(A) = Cn .
8. The columns of A are a basis for Cn .
9. The rank of A is n, r (A) = n.
10. The nullity of A is zero, n (A) = 0.
11. The determinant of A is nonzero, det (A) 6= 0.
12. λ = 0 is not an eigenvalue of A.
13. The linear transformation T : Cn → Cn defined by T (x) = Ax is invertible.
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Definition EELT Eigenvalue and Eigenvector of a Linear Transformation 308
Suppose that T : V → V is a linear transformation. Then a nonzero vector v ∈ V is an eigenvector
of T for the eigenvalue λ if T (v) = λv.
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Definition CBM Change-of-Basis Matrix 309
Suppose that V is a vector space, and IV : V → V is the identity linear transformation on V .
Let B = {v1 , v2 , v3 , . . . , vn } and C be two bases of V . Then the change-of-basis matrix from
B to C is the matrix representation of IV relative to B and C,
IV
CB,C = MB,C
= [ ρC (IV (v1 ))| ρC (IV (v2 ))| ρC (IV (v3 ))| . . . |ρC (IV (vn )) ]
= [ ρC (v1 )| ρC (v2 )| ρC (v3 )| . . . |ρC (vn ) ]
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Theorem CB Change-of-Basis 310
Suppose that v is a vector in the vector space V and B and C are bases of V . Then
ρC (v) = CB,C ρB (v)
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Theorem ICBM Inverse of Change-of-Basis Matrix 311
Suppose that V is a vector space, and B and C are bases of V . Then the change-of-basis matrix
CB,C is nonsingular and
−1
CB,C = CC,B
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Theorem MRCB Matrix Representation and Change of Basis 312
Suppose that T : U → V is a linear transformation, B and C are bases for U , and D and E are
bases for V . Then
T T
MB,D = CE,D MC,E CB,C
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Theorem SCB Similarity and Change of Basis 313
Suppose that T : V → V is a linear transformation and B and C are bases of V . Then
T −1 T
MB,B = CB,C MC,C CB,C
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Theorem EER Eigenvalues, Eigenvectors, Representations 314
Suppose that T : V → V is a linear transformation and B is a basis of V . Then v ∈ V is an
T
eigenvector of T for the eigenvalue λ if and only if ρB (v) is an eigenvector of MB,B for the
eigenvalue λ.
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Definition UTM Upper Triangular Matrix 315
The n × n square matrix A is upper triangular if [A]ij = 0 whenever i > j.
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Definition LTM Lower Triangular Matrix 316
The n × n square matrix A is lower triangular if [A]ij = 0 whenever i < j.
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Theorem PTMT Product of Triangular Matrices is Triangular 317
Suppose that A and B are square matrices of size n that are triangular of the same type. Then
AB is also triangular of that type.
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Theorem ITMT Inverse of a Triangular Matrix is Triangular 318
Suppose that A is a nonsingular matrix of size n that is triangular. Then the inverse of A, A−1 ,
is triangular of the same type. Furthermore, the diagonal entries of A−1 are the reciprocals of
−1
−1
the corresponding diagonal entries of A. More precisely, A ii = [A]ii .
c 2004—2015 Robert A. Beezer, GFDL License
Theorem UTMR Upper Triangular Matrix Representation 319
Suppose that T : V → V is a linear transformation. Then there is a basis B for V such that the
T
matrix representation of T relative to B, MB,B , is an upper triangular matrix. Each diagonal
entry is an eigenvalue of T , and if λ is an eigenvalue of T , then λ occurs αT (λ) times on the
diagonal.
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Theorem OBUTR Orthonormal Basis for Upper Triangular Representation 320
Suppose that A is a square matrix. Then there is a unitary matrix U , and an upper triangular
matrix T , such that
U ∗ AU = T
and T has the eigenvalues of A as the entries of the diagonal.
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Definition NRML Normal Matrix 321
The square matrix A is normal if A∗ A = AA∗ .
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Theorem OD Orthonormal Diagonalization 322
Suppose that A is a square matrix. Then there is a unitary matrix U and a diagonal matrix D,
with diagonal entries equal to the eigenvalues of A, such that U ∗ AU = D if and only if A is a
normal matrix.
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Theorem OBNM Orthonormal Bases and Normal Matrices 323
Suppose that A is a normal matrix of size n. Then there is an orthonormal basis of Cn composed
of eigenvectors of A.
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Definition CNE Complex Number Equality 324
The complex numbers α = a + bi and β = c + di are equal, denoted α = β, if a = c and b = d.
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Definition CNA Complex Number Addition 325
The sum of the complex numbers α = a + bi and β = c + di , denoted α + β, is (a + c) + (b + d)i.
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Definition CNM Complex Number Multiplication 326
The product of the complex numbers α = a+bi and β = c+di , denoted αβ, is (ac−bd)+(ad+bc)i.
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Theorem PCNA Properties of Complex Number Arithmetic 327
The operations of addition and multiplication of complex numbers have the following properties.
• ACCN Additive Closure, Complex Numbers: If α, β ∈ C, then α + β ∈ C.
• MCCN Multiplicative Closure, Complex Numbers: If α, β ∈ C, then αβ ∈ C.
• CACN Commutativity of Addition, Complex Numbers: For any α, β ∈ C, α + β = β + α.
• CMCN Commutativity of Multiplication, Complex Numbers: For any α, β ∈ C, αβ = βα.
• AACN Additive Associativity, Complex Numbers: For any α, β, γ ∈ C, α + (β + γ) =
(α + β) + γ.
• MACN Multiplicative Associativity, Complex Numbers: For any α, β, γ ∈ C, α (βγ) =
(αβ) γ.
• DCN Distributivity, Complex Numbers: For any α, β, γ ∈ C, α(β + γ) = αβ + αγ.
• ZCN Zero, Complex Numbers: There is a complex number 0 = 0+0i so that for any α ∈ C,
0 + α = α.
• OCN One, Complex Numbers: There is a complex number 1 = 1+0i so that for any α ∈ C,
1α = α.
• AICN Additive Inverse, Complex Numbers: For every α ∈ C there exists −α ∈ C so that
α + (−α) = 0.
• MICN Multiplicative
Inverse, Complex Numbers: For every α ∈ C, α 6= 0 there exists
1 1
α ∈ C so that α α = 1.
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Theorem ZPCN Zero Product, Complex Numbers 328
Suppose α ∈ C. Then 0α = 0.
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Theorem ZPZT Zero Product, Zero Terms 329
Suppose α, β ∈ C. Then αβ = 0 if and only if at least one of α = 0 or β = 0.
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Definition CCN Conjugate of a Complex Number 330
The conjugate of the complex number α = a + bi ∈ C is the complex number α = a − bi.
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Theorem CCRA Complex Conjugation Respects Addition 331
Suppose that α and β are complex numbers. Then α + β = α + β.
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Theorem CCRM Complex Conjugation Respects Multiplication 332
Suppose that α and β are complex numbers. Then αβ = αβ.
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Theorem CCT Complex Conjugation Twice 333
Suppose that α is a complex number. Then α = α.
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Definition MCN Modulus of a Complex Number 334
The modulus of the complex number α = a + bi ∈ C, is the nonnegative real number
√ p
|α| = αα = a2 + b2 .
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Definition SET Set 335
A set is an unordered collection of objects. If S is a set and x is an object that is in the set S,
we write x ∈ S. If x is not in S, then we write x 6∈ S. We refer to the objects in a set as its
elements.
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Definition SSET Subset 336
If S and T are two sets, then S is a subset of T , written S ⊆ T if whenever x ∈ S then x ∈ T .
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Definition ES Empty Set 337
The empty set is the set with no elements. It is denoted by ∅.
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Definition SE Set Equality 338
Two sets, S and T , are equal, if S ⊆ T and T ⊆ S. In this case, we write S = T .
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Definition C Cardinality 339
Suppose S is a finite set. Then the number of elements in S is called the cardinality or size of S,
and is denoted |S|.
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Definition SU Set Union 340
Suppose S and T are sets. Then the union of S and T , denoted S ∪ T , is the set whose elements
are those that are elements of S or of T , or both. More formally,
x ∈ S ∪ T if and only if x ∈ S or x ∈ T
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Definition SI Set Intersection 341
Suppose S and T are sets. Then the intersection of S and T , denoted S ∩ T , is the set whose
elements are only those that are elements of S and of T . More formally,
x ∈ S ∩ T if and only if x ∈ S and x ∈ T
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Definition SC Set Complement 342
Suppose S is a set that is a subset of a universal set U . Then the complement of S, denoted S,
is the set whose elements are those that are elements of U and not elements of S. More formally,
x ∈ S if and only if x ∈ U and x 6∈ S
c 2004—2015 Robert A. Beezer, GFDL License