Offline Exercises for Linear Algebra XM511 Lectures 1 12
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1 This document lists the offline exercises for Lectures 1 12 of XM511, which correspond to Chapter 1 of the textbook. These exercises should be be done in the traditional paper and pencil format. The section ( ) numbers refer to the textbook. Offline Exercises for Linear Algebra XM511 Lectures 1 12 Overview These problems are intended to help you understand the concepts covered in lecture, and in particular, to give you experience with writing proofs in linear algebra. If you have difficulty with the proofs in a given section, then it may be helpful to work through some more concrete examples involving that material. These problems provide some such exercises, but you will find many, many more in the textbook. The solutions to these problems are available under Supplemental Course Materials on the course web page. In general, the computational problems will just have answers, but the prooforiented problems will have fully written solutions. Read these carefully, and use them to model the organization and structure of your own proofs. Guidelines for Writing Proofs At the beginning of some problems, you will find a bracketed reference to the Guidelines for Writing Proofs page (found under Supplemental Course Materials/Mathematical Exposition, Writing Proofs), which discusses a number of essential tips and techniques for proof-writing. Whenever you encounter these problems, you should consult the Guidelines page for information on the appropriate technique, and after attempting the problem yourself, be sure that you fully understand the written solution for it. (If you have questions about whether or not your own solution covers the essential elements involved, don t hesitate to ask your instructor!) Here are the items most commonly referred to, and their corresponding topics: [G-II] Mathematical induction; [G-III(4)] Proving if and only if statements; [G-IV] Proofs directly from the definition. Lecture 1 ( 1.1) No problems assigned. Lecture 2 ( 1.1) 1. Determine the orders of the following matrices: [ ] A =, B =, C = D = , E = F = [ ] 1/2 1/3 1/ G =, H = /3 3/5 5/ J = [ ]. 2. Determine which, if any, of the matrices defined in Problem 1 are square. 1
2 3. Determine which, if any, of the matrices defined in Problem 1 are row matrices and which are column matrices. 4. Construct a five-dimensional row matrix having the value j 2 as its jth component. 5. Construct the 2 2 matrix A having a ij = ( 1) i+j. 6. Construct the n n matrix B having b ij = n i j. What will this matrix be when specialized to the 3 3 case? 7. Construct the 3 4 matrix D having i+j, when i > j; d ij = 0, when i = j; i j, when i < j. Lecture 3 ( 1.1) Use the following matrices for problems 1 5: [ ] A =, B =, C = D = , E = Calculate 5A. 2. Calculate 3A 2C. 3. Calculate 0.1A + 0.2C. 4. Find X if 3D X = E. 5. Find R if 4A+5R = 10C. 6. Find 6A θb if A = [ ] θ 2 2θ 1 4 1/θ and B = [ ] θ /θ θ 2. +2θ+1 7. Prove that if 0 is a zero matrix having the same order as A, then A+0 = A. 8. Prove that if A is any matrix and λ 1 and λ 2 are any two scalars, then (λ 1 +λ 2 )A = λ 1 A+λ 2 A. 9. Prove that if A is any matrix and λ 1 and λ 2 are any two scalars, then (λ 1 λ 2 )A = λ 1 (λ 2 A). 2
3 Lecture 4 ( 1.2) 1. In parts (a) (h), calculate the indicated products for: A =, B =, C = D = , E = [ 1 2 ]. 2 2 [ ] (a) AB. (b) BA. (c) AC. (d) BC. (e) EA. (f) AE. (g) ED. (h) DE Find AB for A = and B =. What rule of ordinary multiplication fails to hold here? [ ] Find AB and CB for A =, B = 1 0 multiplication fails to hold here? 4. Calculate the following products: [ ] 1 2 x (a) (b) 3 4][ x y y z [ ] a11 a (c) 12 x b11 b (d) 12 b 13 x y a 21 a 22 y b 21 b 22 b 23 z , and C =. What rule of ordinary The time requirements for a company to produce three products is collected in the matrix T = , where the rows pertain to lamp bases, cabinets, and tables, respectively. The columns pertain to the hours of labor required for cutting the wood, assembling, and painting, respectively. The hourly wages of a carpenter to cut wood, of an artisan to assemble a product, and of a decorator to paint are given, respectively, by the column matrix w = Calculate the product T w, and determine its significance. 6. For parts (a) (c), write the given system in matrix form, Ax = b. (a) 2x + 3y = 10 (b) 2x y = 12 (c) 2x 1 + 3x 2 + 2x 3 + 4x 4 = 5 4x 5y = 11 4y z = 15 x 1 + x 2 + x 4 = 0 3x 1 + 2x 2 + 2x 3 = 3 x 1 + x 2 + 2x 3 + 3x 4 = 4 7. Use the definition of matrix multiplication to prove that jth column of (AB) = A (jth column of B). 3
4 8. Use the definition of matrix multiplication to prove that ith row of (AB) = (ith row of A) B. 9. Provethat if A has a rowof zeros and B is any matrix for which the product AB is defined, then AB also has a row of zeros. 10. Let e j denote the n 1 column matrix with a 1 in its (j,1) position and zeros elsewhere, often called the jth standard basis vector. For example, if n = 3, the standard basis vectors are e 1 = 1 0, e 2 = 0 1, e 3 = Prove that if A is an m n matrix, then Ae j is the jth column of A. Lecture 5 ( 1.3) 1. For each of the following pairs of matrices A and B, find the products (AB) T, A T B T, and B T A T, and verify that (AB) T = B T A T (a) A =, B = [ ] (b) A =, B = (c) A = , B = Find x T y and xy T for each of the following pairs of vectors: (a) x = 2 3, y = 2 3. (b) x = 3 1, y = 1 1. (c) x = 2 0, y = Simplify the following expressions: (a) (AB T ) T (b) (A+B T ) T +A T (c) [A T (B +C T )] T (d) [(AB) T +C] T (e) [(A+A T )(A A T )] T Prove that if A and B are diagonal matrices of the same dimension, then AB = BA. 5. If A is a 2 2 diagonal matrix, then does A necessarily commute with every other 2 2 matrix? Prove your answer. 6. Let A, B, and C be matrices of order m p, p r, and r s, respectively. Prove that (ABC) T = C T B T A T. 7. Prove that if D = [ d ij ] is a diagonal matrix, then D 2 = [ d 2 ij]. 8. (a) Prove that if A is a square matrix, then B = (A+A T )/2 is a symmetric matrix. (b) Prove that if A is a square matrix, then C = (A A T )/2 is a skew-symmetric matrix. (c) Use the results of parts (a) and (b) to prove that any square matrix can be written as the sum of a symmetric matrix and a skew-symmetric matrix.. 4
5 9. A square matrix A is called nilpotent if there exists an integer k such that A k = 0. (a) Prove that if A and B are nilpotent matrices that commute (i.e. AB = BA), then AB is nilpotent. (b) If A and B are nilpotent n n matrices that do not commute, then is AB necessarily nilpotent? If so, prove it. If not, find a counterexample. 10. A square matrix A is called a root of identity if there exists a positive integer k such that A k = I. Prove that if A is a root of identity, then A m is a root of identity for every positive integer m, as well. 11. [G-IV] Suppose that A is a symmetric n n matrix, B is a skew-symmetric n n matrix, and AB +BA = 0. (a) Directly from the definitions of symmetric, skew-symmetric, and matrix multiplication, prove that AB is symmetric. (b) Using Theorem 1 ( 1.3), prove that AB is symmetric. 12. [G-II] Let A 1,A 2,...,A n be matrices of dimensions such that the product A 1 A 2 A n is defined. Using mathematical induction, prove that (A 1 A 2 A n ) T = A T n A T 2A T Prove the following: (a) If a 2 2 matrix A commutes with every 2 2 diagonal matrix, then A must be diagonal. [ ] 1 0 Hint: Consider, in particular, the diagonal matrix D =. 0 0 (b) If an n n matrix A commutes with every n n diagonal matrix, then A must be diagonal. 14. [G-III(4)] Prove that an n n matrix A commutes with every other n n matrix if and only if A = λi for some constant λ. Lecture 6 ( 1.3) Partition A = into block diagonal form and then calculate A Use partitioning to calculate A 2 and A 3 for A =
6 3. Determine which, if any, of the following matrices are in row-reduced form A = , B = C = , D = E = , F = , G = H = , J = , K = L = /2 1/ , M = 0 1 1/4, N = Q =, R =, S =, T = (a) Determine which, if any, of the matrices in Problem 3 are upper triangular. (b) Is a square matrix in row-reduced form necessarily upper triangular? Lecture 7 ( 1.4) 1. Find a value for k such that x = 2 and y = k is a solution of the system 3x + 5y = 11 2x 7y = Find a value for k such that x = 2k, y = k, and z = 0 is a solution of the system x + 2y + z = 0 2x 4y + 2z = 0 3x 6y 4z = Show graphically that the number of solutions to a linear system of two equations in three variables is either none or infinitely many. 4. Let y be a solution to Ax = b, and let z be a solution to the associated homogeneous system Ax = 0. Prove that u = y+z is also a solution to Ax = b. 5. Let y and z be as defined in Problem 4. (a) For what scalars α is u = y+αz also a solution to Ax = b? (b) For what scalars α is u = αy+z also a solution to Ax = b? 6. Let A be an n m matrix, and let b be an n 1 column vector. Prove that if there is a nonzero solution to Ax = 0, then the system Ax = b has either no solutions or infinitely many solutions. 7. [G-III(4)] Let A be an n m matrix, and let b be an n 1 column vector such that the system Ax = b has at least one solution. Prove that this system has exactly one solution if and only if Ax = 0 has only one solution. 6
7 Lecture 8 ( 1.4) In Problems 1 3, write the set of equations associated with the given augmented matrix and the specified variables, and then solve them for x, y, and z for x, y, and z for x 1, x 2, and x Lecture 9 ( 1.4) 1. For parts (a) (c), use Gaussian elimination to solve the given system of equations. (a) y 2z = 4 (b) 3x + 3y 3z = 0 (c) x 1 + 2x 2 x 3 = 1 x + 3y + 2z = 1 x y + 2z = 0 2x 1 3x 2 + 2x 3 = 4 2x + 3y + z = 2 2x 2y + z = 0 x + y + z = 0 2. A mining company has a contract to supply 70,000 tons of low-grade ore, 181,000 tons of medium-grade ore, and 41,000 tons of high-grade ore to a supplier. The company has three mines that it can work. Mine A produces 8,000 tons of low-grade ore, 5,000 tons of medium-grade ore, and 1,000 tons of high-grade ore during each day of operation. Mine B produces 3,000 tons of low-grade ore, 12,000 tons of medium-grade ore, and 3,000 tons of high-grade ore for each day it is in operation. The figures for mine C are 1,000, 10,000, and 2,000, respectively. How many days must each mine operate to meet contractual demands without producing a surplus? Lecture 10 ( 1.5) 1. Determine if any of the following matrices are inverses for M = 1 1/3 1 3 A =, B =, 1/2 1/ C =, D =. 2/3 1/ Show directly that if ad bc 0, then the inverse of A = A 1 = 3. Prove that if A 2 I = 0, then A 1 = A. 1 ad bc [ ] d b. c a [ ] a b is c d [ ] 1 3 : 2 9 7
8 4. Prove that if A is symmetric, then for any nonsingular matrix B of the same dimension, we have (BA 1 ) T (A 1 B T ) 1 = I. 5. Prove that if A is invertible and λ 0, then (λa) 1 = 1 λ A Prove that if A, B, and C are n n nonsingular matrices, then (ABC) 1 = C 1 B 1 A A square matrix A is called orthogonal if A T = A 1. Prove that if A and B are n n orthogonal matrices, then AB is orthogonal as well. [ ] a b 8. [G-III(4)] Let A =. Prove that A is invertible if and only if ad bc 0. c d Note: For the only if direction of this proof, you may want to show that if ad bc = 0, then A is not invertible. To prove that A is not invertible, assume that there exists a matrix B such that AB = I, and show that this leads to a contradiction. Do not try to take A 1 from the if direction and argue that it is undefined when ad bc = [G-III(4)] Let A and B be n n matrices. Prove that AB is nonsingular if and only if A and B are both nonsingular. 10. [G-II,III(4)] Let A 1,...,A k be nonsingular n n matrices. Prove that A 1 A 2 A k is nonsingular if and only if each A i is nonsingular. 11. [G-III(4)] Prove that an n n matrix A is invertible if and only if the system Ax = b has a unique solution for every n 1 column vector b. Hint: Proving one direction of this statement should be easy. For the other direction, think about the systems Ax = e j. Lecture 11 ( 1.5) 1. For parts (a) (f), find elementary matrices that, when multiplied on the right by the given matrix A, will generate the desired result. (a) Interchange the order of the first and second rows of a 2 2 matrix A. (b) Multiply the second row of a 2 2 matrix A by 5. (c) Add to the second row of a 2 2 matrix three times its first row. (d) Add to the second row of a 3 3 matrix A three times its third row. (e) Add to the third row of a 3 4 matrix A five times its first row. (f) Interchange the order of the second and fourth rows of a 6 6 matrix A. 2. For parts (a) (e), find the inverses of the given matrices, if they exist. [ ] 1 1 (a) (b) (c) (d) (e) For parts (a) (c), use matrix inversion, if possible, to solve the given systems of equations: (a) 4x + 2y = 6 (b) 2x + 3y = 8 (c) x + 2y 2z = 1 2x 3y = 1 6x + 9y = 24 2x + y + z = 5 x + y z = 2 8
9 4. If A is a nonsingular matrix, we may define A n = (A 1 ) n for any positive integer n. Use this definition to find A 2 and A 3 for the following matrices: [ ] (a), (b), (c), (d) , (e) Lecture 12 ( 1.6) 1. For parts (a) (c), A and b are given. Construct an LU decomposition for the matrix A, and then use it to solve the system Ax = b for x. (a) A = , b = (b) A = , b = (c) A = , b = (a) Use LU decomposition to solve the system x + 2y = 9 2x + 3y = 4. (b) Use the decomposition to solve the preceding system when the right sides of the equations are replaced by 1 and 1, respectively Solve the system Ax = b for the following vectors b when A = : (a) b = 1 1, (b) b = 0 0, (c) b = , (d) b = Show that LU decomposition cannot be used to solve the system 2y + z = 1 x + y + 3z = 8 2x y z = 1, but that the decomposition can be used if the first two equations are interchanged. 5. [G-III(4),IV] Call a matrix A stable if all of its diagonal elements are nonzero. Suppose that A and B are two lower triangular matrices. (a) Directly from the definitions of lower triangular and matrix multiplication, prove that AB is stable if and only if A and B are both stable. (b) Recallfrom lecturethat a lowertriangularmatrix isinvertible ifand onlyif its diagonal elements are nonzero. Use this to prove that AB is stable if and only if A and B are stable. 9
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