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1 MAT 237 LINEAR ALGEBRA I 2625 Dokuz Eylül University, Faculty of Science, Department of Mathematics web: Instructor: Engin Mermut HOMEWORK 2 MATRIX ALGEBRA Textbook: Linear Algebra: A Geometric Approach, Theodore Shifrin and Malcolm R Adams, 2nd edition, W H Freeman and Company, 2 Course Assistant: For your questions about the problems, see the course assistant: Zübeyir Türkoğlu, Phone: (232) SOLVE ALL EXERCISES FROM YOUR TEXTBOOK Solve all of the exercises in each section to be sure that you have learned the concepts, the precise definitions and you can write proofs of the asked results There are nearly 5 exercises at the end of each section Be sure that you can do the computational exercises without error You must understand the algorithms given in the proofs of the theorems, this will help you understand the theorems and get a feeling why the results of these theorems are true Also understand the geometric examples and the geometric motivation for the intuition behind Section 2 Matrix Operations (pages 89 9): All exercises 5 Section 22 Linear Transformations: An Introduction (pages 2): All exercises 5 Section 23 Inverse Matrices (pages 8 ): All exercises 6 Section 24 Elementary Matrices: Rows Get Equal Time (pages 8 9): All exercises 4 Section 25 The Transpose (pages 2 24): All exercises 24 DEFINITIONS Write the precise definition of the following terms integers Let n, m be positive What are the column vectors and row vectors of an m n matrix? 2 What is a square matrix? 3 What is a diagonal matrix? What is an upper triangular matrix? What is a lower triangular matrix? Are all these matrices square matrices? Is a diagonal matrix also an upper triangular and a lower triangular matrix? 4 What is the zero matrix? Is the zero matrix a square matrix? 5 What is the identity matrix? Is the identity matrix a square matrix? Is it a diagonal matrix? Is it an upper triangular matrix? Is it a lower triangular matrix? 6 When is the sum of two matrices A and B defined? What is the definition of the matrix sum A + B for matrices A and B of appropriate sizes? 7 What is the scalar multiplication of a matrix A by a real number c? ca =? 8 When is the product of two matrices A and B defined? What is the definition of the matrix product AB for matrices A and B of appropriate sizes? 9 What is the projection map P l : R 2 R 2 onto a line l in R 2 through origin? What is the reflection map R l : R 2 R 2 across a line l in R 2 through origin?

2 What is the rotation map T θ : R 2 R 2 of the plane by an angle θ R? 2 For an m n matrix A, what is the function µ A : R n R m that is called the left multiplication by A map? 3 What is the standard matrix of a linear transformation T : R 2 R 2? 4 What is the standard matrix of a linear transformation T : R n R m? 5 Let A be an m n matrix? What is a left inverse of A? What is a right inverse of A? 6 Let A be an n n square matrix When do we say that A is an invertible matrix? What is an inverse of A? 7 What is the algebra of n n matrices over R? Which properties does it have? 8 What is the group GL n (R) (the n n general linear group over R)? 9 What is an m m elementary matrix? What is its usage? 2 What is an LU-decomposition of an m n matrix A? Does it always exist? 2 What is the transpose of an m n matrix A? 22 What is a symmetric n n square matrix A? What is a skew-symmetric n n square matrix A? TRUE OR FALSE Prove your answer You must prove it if it is true If it is false, you must give a counterexample or show why it is false; again a proof is required of course Let m, n, k, p, q Z + The below questions are for the n-dimensional space R n and for matrices whose entries are real numbers The elements of the n-dimensional space R n are written as column vectors: x x 2 x = x n Rn A square matrix that is both upper triangular and lower triangular must be a diagonal matrix 2 (B + C)A = AB + AC for all n n square matrices A, B, C 3 A(BC) = (AB)C for every m n matrix A, n p matrix B and p q matrix C 4 Let A be an m n matrix (a) A(x + y) = Ax + Ay for all x, y R n (b) A(cx) = c(ax) for all x R n and c R (c) A(c v + c 2 v c k v k ) = c (Av ) + c 2 (Av 2 ) + + c k (Av k ) for all v v 2,, v k R n and c, c 2,, c k R 5 Let A be an m n matrix, B be an n p matrix and C be an p q matrix Let A, A 2,, A m be the m rows of A Let c, c 2,, c q be the q columns of the matrix C For each i =, 2,, m and j =, 2,, q: (a) The ith row of AB is A i B (b) (c) (d) The ith row of A(BC) is A i (BC) The ith row of (AB)C is (A i B)C A i (BC) = (A i B)C (e) The jth column of BC is Bc j 2

3 (f) The jth column of A(BC) is A(Bc j ) (g) The jth column of (AB)C is (AB)c j (h) A(Bc j ) = (AB)c j 6 If for matrices A and B, both of the products AB and BA are defined, then both of A and B must be square matrices 7 If A 2 = AA is defined for a matrix A, then A must be a square matrix 8 AB = BA for some n n square matrices A and B 9 If n >, then matrix multiplication in the ring of n n square matrices is a noncommutative operation (ca)b = c(ab) for every m n matrix A, n p matrix B and c R A(cB) = c(ab) for every m n matrix A, n p matrix B and c R 2 AI n = A = I n A for every m n matrix A, where I n is the n n identity matrix 3 The jth column of an m n matrix A is Ae j for each j =, 2,, n, where e j = is the column vector with on the jth row and zeros elsewhere 4 The ith row of an m n matrixa is e T i A for each i =, 2,, m, where e i = is the column vector with on the ith row and zeros elsewhere (and so e T i = [ ] with on the ith column and zeros elsewhere) 5 For any m n matrix A, we denote by µ A : R n R m the multiplication from the left by A function, that is, the function defined by µ A (x) = Ax for all x R n (a) Let A and A be m n matrices, B be an n p matrix Consider the functions (b) (c) µ A : R n R m, µ A : R n R m and µ B : R p R n i µ A µ B = µ AB ii If µ A = µ A, then A = A If A is m n matrix and m < n, then for any b R m, the system Ax = b is either inconsistent or it has infinitely many solutions If A is m n matrix and m > n, then the function µ A : R n R m defined by µ A (x) = Ax for all x R n is onto (d) For any m n matrix A with rank r: i The image Im(µ A ) of µ A consists of all b R m such that the system of equations Ax = b is consistent ii The function µ A is onto if and only if r = n 3

4 iii The function µ A is one-to-one if and only if r = m iv The function µ A is one-to-one if and only if the homogeneous system Ax = has only the trivial solution x = (e) For any n n square matrix A with rank r: i The function µ A is onto if and only if A is nonsingular, that is, r = n ii The function µ A is one-to-one if and only if A is nonsingular, that is, r = n iii The function µ A is onto if and only if for every b R m, the homogeneous system Ax = b has a solution iv The function µ A is onto if and only if for every b R m, the homogeneous system Ax = b has a unique solution v If the function µ A is onto, then µ A is a one-to-one correspondence vi If the function µ A is one-to-one, then µ A is a one-to-one correspondence 6 If T : R n R m and U : R n R m are linear transformations, and c, d R, then the function ct + du : R n R m defined by (ct + du)(x) = ct (x) + du(x) for all x R n is also a linear transformation 7 If T : R n R m and S : R p R n are linear transformations, then their composition T S = T S : R p R n defined by (T S)(x) = (T S)(x) = T (S(x)) for all x R n is also a linear transformation 8 If θ R, the the function T θ : R 2 R 2 that rotates the plane by an angle θ around origin counterclockwise is a linear transformation 9 The standard matrix of the linear[ transformation ] T θ : R 2 R 2 that rotates the plane by an cos θ sin θ angle θ around origin counterclockwise is sin θ cos θ 2 Let l be the line in R 2 through the origin spanned by a nonzero vector a Consider the reflection map R l : R 2 R 2 across the line l in R 2 and the ( projection ) map P l : R 2 R 2 that projects every vector x a in R 2 onto the line l (so P l (x) = proj a x = a 2 a for all x R 2 ) (a) R l is a linear transformation (b) R l = 2P l I where I : R 2 R 2 is the identity map given by I(x) = x for all x R 2 [ ] [ ] cos θ cos 2θ sin 2θ (c) If a =, then the standard matrix of R sin θ l is sin 2θ cos 2θ 2 Every m n matrix A has a right inverse 22 Every m n matrix A has a left inverse 23 Every n n square matrix A is invertible 24 If an m n matrix A has a left inverse C, then for all b R m, the system Ax = b has a unique solution 25 If an m n matrix A has a right inverse B, then for each b R m, the system Ax = b has a unique solution whenever it has at least one solution 26 If an m n matrix A has a right inverse B, then the system Ax = has a unique solution 27 If an m n matrix A with rank r has a right inverse B, then r = n 28 If an m n matrix A has a right inverse B, then A has a unique right inverse 29 If an m n matrix A with rank r has a left inverse C, then r = m 3 If an m n matrix A has a left inverse C, then A has a unique left inverse 4

5 3 If an m n matrix A with rank r has both a right inverse B and a left inverse C, then r = m = n, that is, A is an n n square matrix with rank r = n, and moreover B = C in this case so that AB = I n = BA where I n is the n n identity matrix 32 If an m n matrix A has both a left inverse and a right inverse, then A is an n n square matrix whose reduced echelon form is the n n identity matrix I n 33 If an n n square matrix A is invertible, then it has a unique inverse which is denoted by A 34 If an m n matrix A has a left inverse and m n, then A does not have a right inverse 35 If an m n matrix A has a right inverse and m n, then A does not have a left inverse 36 If n n square matrices A and B are invertible, then AB is also invertible and (AB) = A B 37 If an n n square matrix A is invertible, then its inverse A is also invertible and (A ) = A 38 The n n identity matrix I n is invertible with I n = I n 39 An n n square matrix A is invertible if and only if A is nonsingular (that is, A has rank r = n) 4 An n n square matrix A is invertible if and only if in any echelon form of A, there are exactly n nonzero rows 4 An n n square matrix A is invertible if and only if the reduced echelon form of A is the n n identity matrix I n 42 An n n square matrix A is either invertible or there exists a nonzero n n matrix B such that AB = 43 For an n n square matrix A with rank r, the system Ax = b has a solution for each b R m if and only if r = n 44 For an n n square matrix A with rank r, the system Ax = b has a unique solution for each b R n if and only if r = n 45 If an n n square matrix A has a right inverse B, then B is also a left inverse of A, and so A is invertible with A = B 46 If an n n square matrix A has a left inverse C, then C is also a right inverse of A, and so A is invertible with A = C 47 If A and B are n n matrices that satisfy AB = I n, then A is invertible with A = B 48 If A and B are n n matrices that satisfy AB = I n, then B is invertible with B = A 49 Each elementary m m matrix is obtained by performing the corresponding elementary row operation on the m m identity matrix I m 5 Every elementary matrix is upper triangular 5 There are some elementary matrices that are not invertible 52 Every invertible n n matrix A is a product of finitely many elementary n n matrices 53 The inverse of an elementary matrix is an elementary matrix of the same type 54 If A is an m n matrix with reduced echelon form U, then there exists an invertible matrix P such that P A = U 55 If A is an m n matrix with an echelon form U, then there exists a matrix E that is a product of finitely many elementary matrices such that EA = U 5

6 56 Every m n matrix A has an LU-decomposition 57 The elementary matrix corresponding to the adding a scalar multiple of a row to a row below it is an upper triangular matrix with all diagonal entries 58 The inverse of the elementary matrix corresponding to the adding a scalar multiple of a row to a row below it is a lower triangular matrix with all diagonal entries 59 If A is an m n matrix and U is an echelon form of A obtained by just using elementary row operations that adds a scalar multiple of a row to a row below it, then A = LU for some lower triangular m m invertible matrix with all diagonal entries 6 If A is an n n square matrix that has an echelon form obtained by just using elementary row operations that adds a scalar multiple of a row to a row below it, then A = LU for some lower triangular matrix with all diagonal entries and an upper triangular matrix U 6 If A is an invertible n n square matrix that has an echelon form obtained by just using elementary row operations that adds a scalar multiple of a row to a row below it, then A = LU for some lower triangular matrix with all diagonal entries and an upper triangular matrix U with nonzero diagonal entries 62 (AB) T = A T B T for all m n matrices A and n p matrices B 63 (A T ) T = A for all m n matrices A 64 (ca) T = c(a T ) for all m n matrices A and scalars c R 65 (A + B) T = A T + B T for all m n matrices A and B 66 If A is an invertible n n square matrix, then A T is also invertible and (A T ) = A 67 If an m n matrix A has a right inverse B, then B T is a left inverse of A T 68 If an m n matrix A has a left inverse C, then C T is a right inverse of A T 69 If we identify matrices with real numbers (that is, for all c R, we identify the matrix [c] with the real number c), then the dot product of two vectors in R n is given by: x = x x 2 x n and y = x y = x T y = (the product of the n matrix x T and n matrix y) }{{} dot product in R n 7 (Ax) y = x (A T y) for every m n matrix A and for every }{{}}{{} dot product in R m dot product in R n x = x x 2 x n Rn and y = 7 If A is an n n square matrix that is symmetric, then y y 2 y n y y 2 y m (Ax) y = x (Ay) }{{}}{{} dot product in R n dot product in R n Rm 6

7 for all vectors in R n 72 If a = a a 2 a n x = x x 2 x n and y = y y 2 y n is a nonzero vector in Rn, then for every x = proj a x = ( ) a 2 aat x 73 For an m n matrix A and x R n, if (A T A)x =, then Ax = x x 2 x n in Rn, REVIEW QUESTIONS Be sure that you know the answers to the following questions; review your lecture notes and see your textbook to learn the related concepts: Is R n a vector space over R with the usual addition of vectors in R n and scalar multiplication of vectors by real numbers? Do you know the definition of a vector space over R? Is the set M m n of all m n matrices over R a vector space over R with the matrix addition and scalar multiplication of matrices by real numbers? 2 If A is an m n matrix and B is a n p matrix, express the following: (a) What is the jth column of AB for each j =, 2,, p? Express it as a linear combination of the n columns a, a 2,, a n of A (b) What is the ith row of AB for each i =, 2,, m? Express it as a linear combination of the m rows A, A 2,, A m of A 3 When are the powers A 2 = AA, A 3 = AAA, etc are defined? Must A be a square matrix for these powers to be defined? 4 If A is a diagonal matrix, how do we find the powers A k for each k Z +? 5 Are there n n square matrices such that A k = for some n Z + 6 In terms of the rank r of an m n matrix A, characterize the following properties: (a) For every b R m, the system Ax = b is consistent, that is, it has at least one solution (b) For each b R m, whenever the system Ax = b is consistent, it has a unique solution (c) The homogeneous system Ax = has only the trivial solution x = 7 For n n square matrix A with rank r, are the following properties equivalent: (a) For every b R n, the system Ax = b is consistent, that is, it has at least one solution (b) For each b R n, whenever the system Ax = b is consistent, it has a unique solution (c) The homogeneous system Ax = has only the trivial solution x = (d) For every b R n, the system Ax = b has a unique solution (e) A is nonsingular (f) r = n (g) A is invertible 8 For any m n matrix A, express the meaning of the previous questions in terms of the function µ A : R n R m defined by µ A (x) = Ax for all x R n 7

8 9 Let l be a line in R 2 through origin that is spanned by a nonzero vector a R 2 (a) What is the projection map P l : R 2 R 2 onto the line l? Is it a linear transformation? If so, what is its standard matrix? (b) What is the reflection map P l : R 2 R 2 across the line l? Is it a linear transformation? If so, what is its standard matrix? For θ R, is the function T θ : R 2 R 2 that rotates the plane by an angle θ around origin a linear transformation? If so, what is its standard matrix? Let A be an m n matrix with rank r (a) What is r if A has a right inverse B? Why? (b) What is r if A has a left inverse C? Why? (c) What is r if A has a right inverse B and a right inverse C? Is A a necessarily a square matrix and B = C in this case? 2 How do you determine if a given n n square matrix A is invertible? How do you find the inverse of an n n invertible matrix A? 3 When does an m n matrix A has a right inverse? How do you find all right inverses of an m n matrix A? When is it unique? 4 When does an m n matrix A has a left inverse? How do you find all left inverses of an m n matrix A? When is it unique? 5 How are the row operations on an m n matrix A is recorded by multiplying with elementary matrices? What are the elementary matrices corresponding to each type of row operation? Are elementary matrices always invertible matrices? If so, what are the inverses of each type of elementary matrices? How do they appear when finding the inverse of an n n square matrix? Can we also use them to find a left inverse of an m n matrix? 6 For an m n matrix A, what is the method for finding an m m matrix E such that EA = U is an echelon form of A and E is a product of finitely many elementary m m matrices? Without finding the elementary matrices and taking their product, we can directly find E also in the way we find constraint equations for b R m to have a consistent system Ax = b? 7 What is an LU-decomposition of an m n matrix A? Does it always exists? How do we find it when it exists? 8 What are the properties of the transpose operation on matrices? PROBLEMS Column correspondence property Let A be an m n nonzero matrix and U be a reduced echelon form of A Let a, a 2,, a n be the n columns of A Let u, u 2,, u n be the n columns of U Suppose that the rank of A is r, and so U has r nonzero rows Say the pivot columns of U are the k th, k 2 th,, k r th columns where k < k 2 < < k r n Prove the following results (a) If c a + c 2 a c n a n = for some c, c 2,, c n R, then c u + c 2 u c n u n = That is, if a linear combination of the columns of A equals, then the linear combination of the columns of U with the same coefficients equals, too (b) The converse of part (a) also holds: If c u + c 2 u c n u n = for some c, c 2,, c n R, then c a + c 2 a c n a n = That is, if a linear combination of the columns of U equals, then the linear combination of the columns of A with the same coefficients equals, too 8

9 (c) If for some j {, 2,, n}, the jth column of U is a linear combination of the other columns of U, then the jth column of A is a linear combination of the corresponding columns of A using the same coefficients, and vice versa (d) For each s =, 2,, r, the sth pivot column of U (that is, the k s th column of U) is e s (which is the m column vector that has in the sth row and zeros elsewhere) (e) No pivot column of U is a linear combination of the previous pivot columns (f) No pivot column of U is a linear combination of the other pivot columns (g) Each non-pivot column of U is a linear combination of the previous pivot columns of U where the coefficients of the linear combination are the entries of that non-pivot column of U More precisely, for each j {, 2,, n}, if the jth column u j of U is not a pivot column of U, and there are s pivot columns of U to the left of the jth column of U (that is, k, k 2,, k s are less than j), then u j is a linear combination of the s preceding pivot columns of U (that is, u j is a linear combination of the k th, k 2 th,, k s th columns of U) and the coefficients of the linear combination are the first s entries of the column u j ; furthermore, all the other entries of u j after the first s entries are zeros (h) A column of U is a pivot column if and only if it is nonzero and not a linear combination of the preceding columns of U (i) Using the previous parts, we obtain that for j {, 2,, n}: i If the jth column of U is a pivot column of U, say j = k s for some s {, 2,, r}, then the j = k s th column of A is not a linear combination of the k th, k 2 th,, k s th columns of A ii None of the k th, k 2 th,, k r th columns of A is a linear combination of the others iii If the jth column of U is not a pivot column and if there are s pivot columns of U to the left of the jth column of U (that is, k, k 2,, k s are less than j), then the jth column of A is a linear combination of the k th, k 2 th,, k s th columns of A (which are to the left of the jth column of A) where the coefficients of the linear combination are the entries of the jth column u j of U 2 Uniqueness of a reduced echelon form of a matrix Let us say that an m n matrix A is row-equivalent to an m n matrix B if B can be obtained from A by a finite sequence of elementary row operations Let A, B, C be m n matrices Prove the following: (a) A is row-equivalent to itself (b) If A is row-equivalent to B, then B is row-equivalent to A Hint: The effect of each of the three elementary row operations can be reversed by an elementary row-operation of the same type (c) If A is row-equivalent to B and B is row-equivalent to C, then A is row-equivalent to C Corollary Row-equivalence is an equivalence relation on the set of all m n matrices, so the set of all m n matrices is divided into row-equivalence classes (d) If A is row-equivalent to B and A is row-equivalent to C, then B is row-equivalent to C (e) In each row equivalence class in the set of all m n matrices, there exists exactly one and only one reduced echelon matrix, so reduced echelon matrices form a set of representatives for rowequivalence classes in the set of all m n matrices Hint: Existence is clear since every matrix is row-equivalent to a matrix in reduced echelon form (by the algorithm to reduce a matrix into a reduced echelon form) To prove uniqueness, it must be proved that if two m n matrices in reduced echelon form are row-equivalent, then they are equal, that is if R and R are row-equivalent m n matrices in reduced echelon form, then R = R Do not get frustrated if you cannot prove! Think about what does the reduced echelon form give you for the linear combinations of the columns? What is the relation between the linear combination of columns of A and the linear combination of columns of a reduced echelon form of A? In a reduced echelon form, what is the role of the pivot columns? Can you express each column which is not a pivot column as a linear combination of the pivot columns? How? In A, and in a reduced echelon form of A? This is a nice hard problem to understand something more, think about it! See the previous question for column correspondence property 9

10 (f) By the previous part, a matrix A is row-equivalent to a unique matrix in reduced echelon form, so we can speak of the reduced echelon form of the matrix A (g) We have defined the rank of a matrix A as the number of non-zero rows in any echelon form of A after we have proved that every echelon form of a matrix A has the same number of non-zero rows Since now we know by the above results that a matrix A is row-equivalent to a unique matrix in reduced echelon form, we can also say that the rank of a matrix is the number of the non-zero rows of the reduced echelon form of that matrix (h) Two m n matrices are row-equivalent to one another if and only if they both have the same reduced echelon form, that is, for two m n matrices A, B, let R A be the row-reduced echelon form of A and R B be the row-reduced echelon form of B Then: A is row-equivalent to B if and only if R A = R B Thus uniqueness of a reduced echelon form gives us a way to determine whether two given m n matrices A and B are row-equivalent; just find their reduced echelon forms and see if they are equal or not Without that uniqueness result, observe that this is not an easy problem to answer 3 Prove that the interchange of two rows of a matrix can be accomplished by a finite sequence of elementary row operations of the other two types 4 Find all n n square matrices A such that A commutes with all n n square matrices B, that is, AB = BA for all n n square matrices B 5 Let A = [a ij ] m,n i,j= be an m n matrix and B = [b ij] n,p i,j= satisfy x = x x 2 x p Rp, y = y y 2 y n be an n p matrix Assume that z z 2 Rn, and z = y = b x + b 2 x b p x p y 2 = b 2 x + b 22 x b 2p x p z m Rm, y n = b n x + b n2 x b np x p and z = a y + a 2 y a n y n z 2 = a 2 y + a 22 y a 2n y n z m = a m y + a m2 y a mn y n (a) Express z i for each i =, 2,, m linearly in terms of x, x 2,, x p, that is find the m p matrix C = [c ij ] m,p i,j= such that z = c x + c 2x c px p z 2 = c 2 x + c 22 x c 2p x p z m = c m x + c m2 x c mp x p Express each c ij in terms of a ij s and b ij s Hint: For each i =, 2,, m and k =, 2, n, we are given that n p z i = a ik y k and y k = b kj x j k= Play with these sums to find each c ij for i =, 2, m and j =, 2,, p j=

11 (b) Show that C is the matrix product AB, that is C = AB That is the reason why the matrix product have been defined in the way we have seen (c) All the above work is nothing but gives the associativity in the following matrix product: y = Bx and z = Ay implies z = Ay = A(Bx) = (AB)x, where x = [x x 2 x p ] T, y = [y y 2 y n ] T and z = [z z 2 z m ] T are column vectors (d) Using this last result, prove the associativity of the matrix product, that is, prove that A(BC) = (AB)C for every m n matrix A, n p matrix B and p q matrix C 6 Prove that the interchange of two rows of a matrix can be accomplished by a finite sequence of elementary row operations of the other two types 7 List all types of (a) 2 3 reduced echelon matrices (b) 3 4 reduced echelon matrices 8 Find a 2 2 matrix A with real entries such that A 2 = I 2, where I 2 is the 2 2 identity matrix 9 Let A, B be m n matrices If for all n matrices X (ie column vectors), AX = BX, then prove that A = B Block multiplication (a) Let M, M be m n and n p matrices, and let r be a positive integer less than n We may decompose the two matrices into blocks as follows: [ ] M = [A B] and M A =, where A has r columns and A has r rows Then show that the matrix product can be computed as follows: MM = AA + BB For example show that, [ 5 7 ] [ ] [ ] 2 3 = B [ 5 7 ] [ ] [ ] 2 3 = 4 8 (b) Suppose for positive integers r, s, k, l, r + s = n and k + l = m; we decompose an m n matrix M and an n p matrix M into submatrices [ ] [ A B M =, M A B ] = C D C D, where the number r of columns of A equals to the number of rows of A (and the number of rows of A is k) Then prove that [ ] [ A B A B ] [ AA + BC AB + BD C D C D = CA + DC CB + DD (c) Generalize this block multiplication rule when the matrices are divided into an arbitrary number of submatrices by partitioning their entries using horizontal and/or vertical lines, of course for the block multiplication rule to be meaningful the necessary submatrix products must be defined, ie some row numbers and column numbers must agree Observe that in this case the product of block matrices is calculated by multiplying the submatrices as if they were scalars (d) Suppose A is an invertible m m square matrix and B is an invertible n n square matrix ]

12 i Show that the matrix [ A B is invertible, and give a formula for its inverse ii Suppose C is an arbitrary m n matrix Is the matrix [ ] A C invertible? B m n matrix units For i =,, m and j =,, n, let E ij be the m n matrix that has a at the (i, j) position as its only nonzero entry, ie all its other entries are zero These mn matrices are called the matrix units Let A = [a ij ] m,n i,j= be an m n matrix ] (a) Show that A = i,j a ij E ij (b) Compute the matrix product E ij AE kl if m = n and i, j, k, l {,, n} (c) Compute E ii AE jj, E ij A and AE ij if m = n and i, j {,, n} 2 For each θ R, the rotation map T θ : R 2 R 2 is the map that rotates the plane around the origin by θ radians counterclockwise (a) Observe geometrically that T θ : R 2 R 2 is a linear transformation (b) What is the standard matrix of the linear transformation T θ : R 2 R 2? (c) Using the standard matrix of the rotation transformation, prove that for all θ, φ R, T θ T φ = T θ+φ (d) Can you obtain the previous result geometrically easily? (e) Prove that for all θ, φ R, T θ T φ = T φ T θ What does this give you for their standard matrices? 3 (a) Prove that the reflection map R l : R 2 R 2 across a line l in R 2 through origin with direction vector a is given for all x R 2 by ( x a ) R l (x) = 2 a x a a Remember that the projection map P l : R 2 R 2 onto the line l is given for all x R 2 by ( x a ) P l (x) = proj a x = a a a So you must show that R l = 2P l I, where I : R 2 R 2 is the identity map given by I(x) = x for all x R 2 because the above formulae say that R l (x) = 2P l (x) x = 2P l (x) I(x) = (2P l I)(x) for all x R 2 (b) Prove that the reflection map R l : R 2 R 2 across a line l in R 2 through origin with normal vector b is given for all x R 2 by ( ) x b R l (x) = x 2 b b b (c) Prove that the reflection map R S : R 3 R 3 across a plane S in R 3 through origin with normal vector b is given for all x R 3 by ( ) x b R S (x) = x 2 proj b (x) = x 2 b b b (d) Can you generalize the last result to the reflection map R H : R n R n across a hyperplane H in R n through origin with normal vector b? 2

13 4 If an n n square matrix A is strictly upper triangular, that is, if all entries of A on the diagonal and below the diagonal are zero, then prove that A n = 5 An n n square matrix A is called nilpotent if A k = for some positive integer k Prove that if A is nilpotent, then I n A and I n + A are invertible, where I n is the n n identity matrix 6 Denote by GL n (R) the set of all invertible n n square matrices Prove that GL n (R) is a group under matrix multiplication 7 Prove that if a product AB of two n n square matrices is invertible, then so are the factors A, B 8 Prove that every invertible 2 2 matrix is a product of at most four elementary matrices [ ] 2 9 Write the matrix as a product of elementary matrices, using as few as you can Prove that 3 4 your expression is as short as possible 2 Trace of square matrices (a) Let A be an n n matrix The trace of A is defined to be the sum of the diagonal elements of A: trace(a) = a + a 22 + a a nn = (b) If A and B are n n square matrices, prove that trace(ab) = trace(ba) n a kk (c) Let A, B be square matrices Prove that if B is invertible, then trace(a) = trace(bab ) (d) Prove that AB BA = I n has no solutions for n n square matrices A, B with real entries, where I n is the n n identity matrix [ ] c c (e) Let C = 2 be a 2 2 matrix over R We inquire when it is possible to find 2 2 c 2 c 22 matrices A and B such that C = AB BA Prove that such matrices can be found if and only if trace(c) =, ie c + c 22 = 2 Let A be an n n square matrix Prove that: (a) If A is invertible and AB = for some n n matrix B, then B = (b) If A is not invertible, then there exists an n n matrix B such that AB = but B 22 Let A be an m n matrix and B be an n m matrix Let I m be the m m identity matrix and I n be the n n identity matrix Prove that I m AB is invertible if and only if I n BA is invertible [ ] a b 23 Let A = Prove using elementary row operations that A is invertible if and only if ad bc c d What is A if A is invertible? 24 Prove that an upper triangular n n square matrix A is invertible if and only if every entry on its diagonal is different from, that is, if and only if a kk for all k =, 2,, n Does the same result hold also for a lower triangular matrix? 25 Prove that if A is an m n matrix and B is an n m matrix and n < m, then the m m square matrix AB is not invertible 26 Permutation matrices An n n square matrix A is said to be a permutation matrix if it is obtained by permuting the rows of the n n identity matrix I n, or equivalently if it is obtained by permuting the columns of the n n identity matrix I n So an n n square matrix A is a permutation matrix if and only if it has a single in each row and in each column, and all its other entries are k= 3

14 For each j =, 2,, n, let e j = be the column vector with on the jth row and zeros elsewhere Let σ : {, 2,, n} {, 2,, n} be a permutation, that is, a one-to-one and onto function Then its inverse function σ : {, 2,, n} {, 2,, n} also exists The permutation matrix P σ associated to the permutation σ is the n n matrix whose jth column is e σ(j) for j =, 2,, n: Observe that P σ = P σ = e σ() e σ(2) e σ(n) e T σ () e T σ (2) e T σ (n) (a) What is P σ A for an n n matrix A? Express in terms of the permutation of the rows of A x x 2 (b) What is P σ? x n (c) What is AP σ for an n n matrix A? Express in terms of the permutation of the columns of A (d) Is a permutation matrix P σ invertible? If so, what is P σ? 27 For an m n matrix A with rank r, prove the following: (a) If r < m, then there exists b R m such that the system Ax = b has no solution (b) The system Ax = b is consistent for all b R m if and only if r = m (c) For each b R m, the system Ax = b has a unique solution whenever it has a solution if and only if r = n (d) The homogeneous system Ax = has the only the trivial solution x = if and only if r = n (e) For all b R m, the system Ax = b has a unique solution if and only if r = m = n 28 For an m n matrix A, if m < n, then the homogeneous system Ax = has infinitely many solutions, and so there exists infinitely many nontrivial solutions (that is solutions different from the trivial solution x = ) On other words, if in a system of homogeneous linear equations, there are more variables than equations (that is, the number n of the variables x, x 2,, x n, is more than the number m of equations in the system Ax = ), then there are infinitely many nontrivial solutions (that is solutions different from the trivial solution x = ) 29 Suppose that an m n matrix A has a right inverse B (a) Prove that for every b R m, x = Bb is a solution of Ax = b (b) If r is the rank of A, then prove that r = m must hold 3 Let A be an m n matrix and C be an n m matrix 4

15 (a) If for every b R m, x = Cb is a solution of Ax = b, then prove that AC = I m where I m is the m m identity matrix (b) If C is a left inverse of A, then prove that the system Ax = b has a unique solution whenever it has a solution (c) If C is a left inverse of A, then prove that the system Ax = has only the trivial solution x = (d) If C is a left inverse of A and if for every b R m, x = Cb is a solution of Ax = b, then prove that m = n (that is, A is a square matrix) and AC = I n = CA which means that the n n square matrix C is the inverse of the n n square matrix A 3 If an m n matrix A has rank r = m, then prove that it has a right inverse B How do we find all right inverses of A in this case? 32 If an m n matrix A has rank r = n, then prove the following: (a) m n must hold and the reduced echelon form of A must be U = (b) There exists elementary matrices E, E 2,, E k such that for E = E k E 2 E, we have EA = U (c) The n m matrix C obtained by taking the first n rows of the m m matrix E is a left inverse of A 33 If an m n matrix A with rank r has a unique right inverse B, then prove that r = m = n, that is, A is an n n square matrix with rank r = n (ie A is nonsingular), and B is the inverse of A, that is, AB = I n = BA where I n is the n n identity matrix 34 If an m n matrix A with rank r has a unique left inverse C, then prove that r = m = n, that is, A is an n n square matrix with rank r = n (ie A is nonsingular), and C is the inverse of A, that is, AC = I n = CA where I n is the n n identity matrix 35 Let A be an m n matrix (a) How do we find all right inverses of A? (b) If C is a left inverse of A, then prove that C T is a right inverse of A T [ In (c) Use the previous two parts to give a method to find all left inverses of A [ ] 36 Prove that the matrix A = has no LU-decomposition 37 If an m n matrix A has an LU-decomposition, is it unique? If it is unique, prove that; otherwise, give a counterexample 38 If A is an n n square matrix and if it has an LU-decomposition, is the LU-decomposition unique? If it is unique, prove that; otherwise, give a counterexample 39 If A is an invertible n n square matrix and if it has an LU-decomposition, then prove that the LU-decomposition of A is unique by showing the following: If L and L are lower triangular matrices with s on the diagonal and U and U are upper triangular matrices with a nonzero diagonal, and if LU = L U, then prove that L = L and U = U Hint: L, L, U, U are invertible matrices; L L = U U where the left side is lower triangular with all diagonal entries and the right side is upper triangular What does this imply? 4 For any m n matrix A, there is an m m permutation matrix P so that P A does have an LUdecomposition ] 5

16 4 Orthogonal matrices (a) Let A be an n n square matrix with columns a, a 2,, a n and rows A, A 2,, A n : A = [a ij ] m,n i,j= = a a 2 a n = A A 2 A n We say that A is an orthogonal matrix if A T A = I n where I n is the n n identity matrix Prove that the following are equivalent for A: i A is an orthogonal matrix, that is, A T A = I n ii A is an invertible matrix with A = A T iii AA T = I n iv The column vectors a, a 2,, a n are unit vectors that are orthogonal to one another, that is, for all i, j {, 2,, n}, {, if i j; a i a j = δ ij =, if i = j v The row vectors A, A 2,, A n are unit vectors that are orthogonal to one another, that is, for all i, j {, 2,, n}, {, if i j; A i A j = δ ij =, if i = j (b) If A and B are orthogonal n n matrices, then prove that AB and A are also orthogonal matrices (c) Denote by O n (R) the set of all n n orthogonal matrices: O n (R) = {A A is an n n orthogonal matrix with real entries} Prove that O n (R) is a subgroup of the n n general linear group GL n (R) (under multiplication of matrices) (d) Show that any 2 2 orthogonal matrix A must be of the form [ ] [ ] cos θ sin θ cos θ sin θ or sin θ cos θ sin θ cos θ for some real number θ (e) If A is a 2 2 orthogonal matrix, then show that the multiplication from the left by A map µ A : R 2 R 2 is either a rotation or the composition of a rotation and a reflection 6

17 SAMPLE QUIZ PROBLEMS Let θ R Find the standard matrix [ of] the reflection map R l : R 2 R 2 across the line l through cos θ the origin in R 2 spanned by a = sin θ 2 Let θ R Find the standard matrix of the rotation map T θ : R 2 R 2 which rotates the plane around the origin by θ radians counterclockwise 3 Let l be the line 3x + 2x 2 = in R 2 [ ] a (a) Find a vector a = that spans the line l so that a 2 l = {ta t R} (b) Find the standard matrix of the projection map P l : R 2 R 2 onto the line l (c) Find the standard matrix of the reflection map R l : R 2 R 2 across the line l 4 Is the following 4 4 matrix A invertible? If so, find its inverse A, and express A and A as a product of elementary matrices Suppose that by making five elementary row operations on the 4 4 matrix A, we obtain the 4 4 identity matrix I 4 These five elementary row operations corresponds to the below five elementary matrices E, E 2, E 3, E 4, E 5, that is, E 5 E 4 E 3 E 2 E A = I 4, where: E = E 2 = E 3 = E 4 = E 5 = means the row operation means the row operation means the row operation means the row operation means the row operation and so E =??? = and so E2 =??? = and so E3 =??? = and so E4 =??? = and so E5 =??? = (a) What is A? Hint: You do not know A but you know the elementary row operations needed to obtain the reduced echelon form of A since E 5 E 4 E 3 E 2 E A = I 4 What do you obtain if you apply these elementary row operations in the same order to the 4 4 identity matrix I 4? 7

18 (b) Find the inverse of each of the elementary matrices E, E 2, E 3, E 4, E 5 just by applying the inverse of the corresponding elementary row operations to the 4 4 identity matrix I 4 (c) Find A by applying the inverse of these elementary row operations in the reverse order to the 4 4 identity matrix I 4, that is, you shall find A = E E 2 E 3 E 4 E 5 I 4 = E E 2 E 3 E 4 E 5 by making the corresponding row operations in this product 8

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