**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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Definition ESYS Equivalent Systems 4 Two systems of linear equations are equivalent if their solution sets are equal. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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]. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem PSSLS Possible Solution Sets for Linear Systems 26 A system of linear equations has no solutions, a unique solution or infinitely many solutions. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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}. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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} c 2004—2015 Robert A. Beezer, GFDL License 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). c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License Theorem CRVA Conjugation Respects Vector Addition 62 Suppose x and y are two vectors from Cm . Then x+y =x+y c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License Theorem IPAC Inner Product is Anti-Commutative 67 Suppose that u and v are vectors in Cm . Then hu, vi = hv, ui. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License Theorem IPN Inner Products and Norms 69 2 Suppose that u is a vector in Cm . Then kuk = hu, ui. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License Theorem OSLI Orthogonal Sets are Linearly Independent 74 Suppose that S is an orthogonal set of nonzero vectors. Then S is linearly independent. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Definition SYM Symmetric Matrix 84 The matrix A is symmetric if A = At . c 2004—2015 Robert A. Beezer, GFDL License Theorem SMS Symmetric Matrices are Square 85 Suppose that A is a symmetric matrix. Then A is square. c 2004—2015 Robert A. Beezer, GFDL License Theorem TMA Transpose and Matrix Addition 86 Suppose that A and B are m × n matrices. Then (A + B)t = At + B t . c 2004—2015 Robert A. Beezer, GFDL License Theorem TMSM Transpose and Matrix Scalar Multiplication 87 Suppose that α ∈ C and A is an m × n matrix. Then (αA)t = αAt . c 2004—2015 Robert A. Beezer, GFDL License Theorem TT Transpose of a Transpose 88 t Suppose that A is an m × n matrix. Then (At ) = A. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License Theorem CRMA Conjugation Respects Matrix Addition 90 Suppose that A and B are m × n matrices. Then A + B = A + B. c 2004—2015 Robert A. Beezer, GFDL License Theorem CRMSM Conjugation Respects Matrix Scalar Multiplication 91 Suppose that α ∈ C and A is an m × n matrix. Then αA = αA. c 2004—2015 Robert A. Beezer, GFDL License Theorem CCM Conjugate of the Conjugate of a Matrix 92 Suppose that A is an m × n matrix. Then A = A. c 2004—2015 Robert A. Beezer, GFDL License Theorem MCT Matrix Conjugation and Transposes 93 t Suppose that A is an m × n matrix. Then (At ) = A . c 2004—2015 Robert A. Beezer, GFDL License Definition A Adjoint 94 t If A is a matrix, then its adjoint is A∗ = A . c 2004—2015 Robert A. Beezer, GFDL License Theorem AMA Adjoint and Matrix Addition 95 ∗ Suppose A and B are matrices of the same size. Then (A + B) = A∗ + B ∗ . c 2004—2015 Robert A. Beezer, GFDL License Theorem AMSM Adjoint and Matrix Scalar Multiplication 96 ∗ Suppose α ∈ C is a scalar and A is a matrix. Then (αA) = αA∗ . c 2004—2015 Robert A. Beezer, GFDL License Theorem AA Adjoint of an Adjoint 97 ∗ Suppose that A is a matrix. Then (A∗ ) = A. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 ] . c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License Theorem MMIM Matrix Multiplication and Identity Matrix 104 Suppose A is an m × n matrix. Then 1. AIn = A 2. Im A = A c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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∗ . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Definition HM Hermitian Matrix 113 The square matrix A is Hermitian (or self-adjoint) if A = A∗ . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License Theorem MIU Matrix Inverse is Unique 118 Suppose the square matrix A has an inverse. Then A−1 is unique. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem MIT Matrix Inverse of a Transpose 121 Suppose A is an invertible matrix. Then At is invertible and (At )−1 = (A−1 )t . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License Theorem NI Nonsingularity is Invertibility 125 Suppose that A is a square matrix. Then A is nonsingular if and only if A is invertible. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 ∗ . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 ). c 2004—2015 Robert A. Beezer, GFDL License Theorem REMRS Row-Equivalent Matrices have equal Row Spaces 138 Suppose A and B are row-equivalent matrices. Then R(A) = R(B). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem CSRST Column Space, Row Space, Transpose 140 Suppose A is a matrix. Then C(A) = R(At ). c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem ZVU Zero Vector is Unique 146 Suppose that V is a vector space. The zero vector, 0, is unique. c 2004—2015 Robert A. Beezer, GFDL License Theorem AIU Additive Inverses are Unique 147 Suppose that V is a vector space. For each u ∈ V , the additive inverse, −u, is unique. c 2004—2015 Robert A. Beezer, GFDL License Theorem ZSSM Zero Scalar in Scalar Multiplication 148 Suppose that V is a vector space and u ∈ V . Then 0u = 0. c 2004—2015 Robert A. Beezer, GFDL License Theorem ZVSM Zero Vector in Scalar Multiplication 149 Suppose that V is a vector space and α ∈ C. Then α0 = 0. c 2004—2015 Robert A. Beezer, GFDL License Theorem AISM Additive Inverses from Scalar Multiplication 150 Suppose that V is a vector space and u ∈ V . Then −u = (−1)u. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License Definition TS Trivial Subspaces 154 Given the vector space V , the subspaces V and {0} are each called a trivial subspace. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem DCM Dimension of Cm 175 The dimension of Cm (Example VSCV) is m. c 2004—2015 Robert A. Beezer, GFDL License Theorem DP Dimension of Pn 176 The dimension of Pn (Example VSP) is n + 1. c 2004—2015 Robert A. Beezer, GFDL License Theorem DM Dimension of Mmn 177 The dimension of Mmn (Example VSM) is mn. c 2004—2015 Robert A. Beezer, GFDL License 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)). c 2004—2015 Robert A. Beezer, GFDL License 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)). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem RPNC Rank Plus Nullity is Columns 181 Suppose that A is an m × n matrix. Then r (A) + n (A) = n. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 ). c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 ). c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem EMN Elementary Matrices are Nonsingular 192 If E is an elementary matrix, then E is nonsingular. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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)) c 2004—2015 Robert A. Beezer, GFDL License Theorem DMST Determinant of Matrices of Size Two 196 a b Suppose that A = . Then det (A) = ad − bc. c d c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem DT Determinant of the Transpose 198 Suppose that A is a square matrix. Then det (At ) = det (A). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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) c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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). c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License Theorem EMHE Every Matrix Has an Eigenvalue 212 Suppose A is a square matrix. Then A has at least one eigenvalue. c 2004—2015 Robert A. Beezer, GFDL License 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 ) c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 ) c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Definition DZM Diagonalizable Matrix 239 Suppose A is a square matrix. Then A is diagonalizable if A is similar to a diagonal matrix. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem DED Distinct Eigenvalues implies Diagonalizable 242 Suppose A is a square matrix of size n with n distinct eigenvalues. Then A is diagonalizable. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License Theorem LTTZZ Linear Transformations Take Zero to Zero 244 Suppose T : U → V is a linear transformation. Then T (0) = 0. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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} c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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)) c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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} c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 ) c 2004—2015 Robert A. Beezer, GFDL License 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}. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License Theorem ILTD Injective Linear Transformations and Dimension 264 Suppose that T : U → V is an injective linear transformation. Then dim (U ) ≤ dim (V ). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 } c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 ). c 2004—2015 Robert A. Beezer, GFDL License 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= ∅ c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License Theorem SLTD Surjective Linear Transformations and Dimension 273 Suppose that T : U → V is a surjective linear transformation. Then dim (U ) ≥ dim (V ). c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Definition IDLT Identity Linear Transformation 275 The identity linear transformation on the vector space W is defined as IW : W → W, IW (w) = w c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License Theorem IVSED Isomorphic Vector Spaces have Equal Dimension 283 Suppose U and V are isomorphic vector spaces. Then dim (U ) = dim (V ). c 2004—2015 Robert A. Beezer, GFDL License 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 )) c 2004—2015 Robert A. Beezer, GFDL License 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 )) c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 ) c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License Theorem VRLT Vector Representation is a Linear Transformation 290 The function ρB (Definition VR) is a linear transformation. c 2004—2015 Robert A. Beezer, GFDL License Theorem VRI Vector Representation is Injective 291 The function ρB (Definition VR) is an injective linear transformation. c 2004—2015 Robert A. Beezer, GFDL License Theorem VRS Vector Representation is Surjective 292 The function ρB (Definition VR) is a surjective linear transformation. c 2004—2015 Robert A. Beezer, GFDL License Theorem VRILT Vector Representation is an Invertible Linear Transformation293 The function ρB (Definition VR) is an invertible linear transformation. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 ). c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 )) ] c 2004—2015 Robert A. Beezer, GFDL License 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)) c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 ) ] c 2004—2015 Robert A. Beezer, GFDL License 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) c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Definition UTM Upper Triangular Matrix 315 The n × n square matrix A is upper triangular if [A]ij = 0 whenever i > j. c 2004—2015 Robert A. Beezer, GFDL License Definition LTM Lower Triangular Matrix 316 The n × n square matrix A is lower triangular if [A]ij = 0 whenever i < j. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Definition NRML Normal Matrix 321 The square matrix A is normal if A∗ A = AA∗ . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem ZPCN Zero Product, Complex Numbers 328 Suppose α ∈ C. Then 0α = 0. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License Theorem CCRA Complex Conjugation Respects Addition 331 Suppose that α and β are complex numbers. Then α + β = α + β. c 2004—2015 Robert A. Beezer, GFDL License Theorem CCRM Complex Conjugation Respects Multiplication 332 Suppose that α and β are complex numbers. Then αβ = αβ. c 2004—2015 Robert A. Beezer, GFDL License Theorem CCT Complex Conjugation Twice 333 Suppose that α is a complex number. Then α = α. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License Definition ES Empty Set 337 The empty set is the set with no elements. It is denoted by ∅. c 2004—2015 Robert A. Beezer, GFDL License 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 . c 2004—2015 Robert A. Beezer, GFDL License 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|. c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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 c 2004—2015 Robert A. Beezer, GFDL License 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