Linear Algebra With Applications
5th Edition
ISBN: 9780321796943
Author: BRETSCHER, Otto.
Publisher: Pearson Education
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Chapter 8.3, Problem 31E
Show that any matrix of rank r can be written as the sum of r matrices of rank 1.
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Let the universal set be whole numbers 1
through 20 inclusive. That is,
U = {1, 2, 3, 4, . . ., 19, 20}. Let A, B, and C
be subsets of U.
Let A be the set of all prime numbers:
A = {2, 3, 5, 7, 11, 13, 17, 19}
Let B be the set of all odd numbers:
B = {1,3,5,7, . . ., 17, 19}
Let C be the set of all square numbers:
C = {1,4,9,16}
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Chapter 8 Solutions
Linear Algebra With Applications
Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - For each of the matrices A in Exercises 7 through...
Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - Let L from R3 to R3 be the reflection about the...Ch. 8.1 - Consider a symmetric 33 matrix A with A2=I3 . Is...Ch. 8.1 - In Example 3 of this section, we diagonalized the...Ch. 8.1 - If A is invertible and orthogonally...Ch. 8.1 - Find the eigenvalues of the matrix...Ch. 8.1 - Use the approach of Exercise 16 to find the...Ch. 8.1 - Consider unit vector v1,...,vn in Rn such that the...Ch. 8.1 - Consider a linear transformation L from Rm to Rn ....Ch. 8.1 - Consider a linear transformation T from Rm to Rn ,...Ch. 8.1 - Consider a symmetric 33 matrix A with eigenvalues...Ch. 8.1 - Consider the matrix A=[0200k0200k0200k0] , where k...Ch. 8.1 - If an nn matrix A is both symmetric and...Ch. 8.1 - Consider the matrix A=[0001001001001000] . Find an...Ch. 8.1 - Consider the matrix [0000100010001000100010000] ....Ch. 8.1 - Let Jn be the nn matrix with all ones on the...Ch. 8.1 - Diagonalize the nn matrix (All ones along both...Ch. 8.1 - Diagonalize the 1313 matrix (All ones in the last...Ch. 8.1 - Consider a symmetric matrix A. If the vector v is...Ch. 8.1 - Consider an orthogonal matrix R whose first column...Ch. 8.1 - True or false? If A is a symmetric matrix, then...Ch. 8.1 - Consider the nn matrix with all ones on the main...Ch. 8.1 - For which angles(s) can you find three distinct...Ch. 8.1 - For which angles(s) can you find four distinct...Ch. 8.1 - Consider n+1 distinct unit vectors in Rn such that...Ch. 8.1 - Consider a symmetric nn matrix A with A2=A . Is...Ch. 8.1 - If A is any symmetric 22 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 22 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 33 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 33 matrix with eigenvalues...Ch. 8.1 - Show that for every symmetric nn matrix A, there...Ch. 8.1 - Find a symmetric 22 matrix B such that...Ch. 8.1 - For A=[ 2 11 11 11 2 11 11 11 2 ] find a nonzero...Ch. 8.1 - Consider an invertible symmetric nn matrix A. When...Ch. 8.1 - We say that an nnmatrix A is triangulizable if A...Ch. 8.1 - a. Consider a complex upper triangular nnmatrix U...Ch. 8.1 - Let us first introduce two notations. For a...Ch. 8.1 - Let U0 be a real upper triangular nn matrix with...Ch. 8.1 - Let R be a complex upper triangular nnmatrix with...Ch. 8.1 - Let A be a complex nnmatrix that ||1 for all...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - If A is a symmetric matrix, what can you say about...Ch. 8.2 - Recall that a real square matrix A is called skew...Ch. 8.2 - Consider a quadratic form q(x)=xAx on n and a...Ch. 8.2 - If A is an invertible symmetric matrix, what is...Ch. 8.2 - Show that a quadratic form q(x)=xAx of two...Ch. 8.2 - Show that the diagonal elements of a positive...Ch. 8.2 - Consider a 22 matrix A=[abbc] , where a and det A...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - a. Sketch the following three surfaces:...Ch. 8.2 - On the surface x12+x22x32+10x1x3=1 find the two...Ch. 8.2 - Prob. 23ECh. 8.2 - Consider a quadratic form q(x)=xAx Where A is a...Ch. 8.2 - Prob. 25ECh. 8.2 - Prob. 26ECh. 8.2 - Consider a quadratic form q(x)=xAx , where A is a...Ch. 8.2 - Show that any positive definite nnmatrix A can be...Ch. 8.2 - For the matrix A=[8225] , write A=BBT as discussed...Ch. 8.2 - Show that any positive definite matrix A can be...Ch. 8.2 - Prob. 31ECh. 8.2 - Prob. 32ECh. 8.2 - Prob. 33ECh. 8.2 - Prob. 34ECh. 8.2 - Prob. 35ECh. 8.2 - Prob. 36ECh. 8.2 - Prob. 37ECh. 8.2 - Prob. 38ECh. 8.2 - Prob. 39ECh. 8.2 - Prob. 40ECh. 8.2 - Prob. 41ECh. 8.2 - Prob. 42ECh. 8.2 - Prob. 43ECh. 8.2 - Prob. 44ECh. 8.2 - Prob. 45ECh. 8.2 - Prob. 46ECh. 8.2 - Prob. 47ECh. 8.2 - Prob. 48ECh. 8.2 - Prob. 49ECh. 8.2 - Prob. 50ECh. 8.2 - What are the signs of the determinants of the...Ch. 8.2 - Consider a quadratic form q. If A is a symmetric...Ch. 8.2 - Consider a quadratic form q(x1,...,xn) with...Ch. 8.2 - If A is a positive semidefinite matrix with a11=0...Ch. 8.2 - Prob. 55ECh. 8.2 - Prob. 56ECh. 8.2 - Prob. 57ECh. 8.2 - Prob. 58ECh. 8.2 - Prob. 59ECh. 8.2 - Prob. 60ECh. 8.2 - Prob. 61ECh. 8.2 - Prob. 62ECh. 8.2 - Prob. 63ECh. 8.2 - Prob. 64ECh. 8.2 - Prob. 65ECh. 8.2 - Prob. 66ECh. 8.2 - Prob. 67ECh. 8.2 - Prob. 68ECh. 8.2 - Prob. 69ECh. 8.2 - Prob. 70ECh. 8.2 - Prob. 71ECh. 8.3 - Find the singular values of A=[1002] .Ch. 8.3 - Let A be an orthogonal 22 matrix. Use the image of...Ch. 8.3 - Let A be an orthogonal nn matrix. Find the...Ch. 8.3 - Find the singular values of A=[1101] .Ch. 8.3 - Find the singular values of A=[pqqp] . Explain...Ch. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - If A is an invertible 22 matrix, what is the...Ch. 8.3 - If A is an invertible nn matrix, what is the...Ch. 8.3 - Consider an nm matrix A with rank(A)=m , and a...Ch. 8.3 - Prob. 18ECh. 8.3 - Prob. 19ECh. 8.3 - Prob. 20ECh. 8.3 - Prob. 21ECh. 8.3 - Consider the standard matrix A representing the...Ch. 8.3 - Consider an SVD A=UVT of an nm matrix A. Show that...Ch. 8.3 - If A is a symmetric nn matrix, what is the...Ch. 8.3 - Prob. 25ECh. 8.3 - Prob. 26ECh. 8.3 - Prob. 27ECh. 8.3 - Prob. 28ECh. 8.3 - Prob. 29ECh. 8.3 - Prob. 30ECh. 8.3 - Show that any matrix of rank r can be written as...Ch. 8.3 - Prob. 32ECh. 8.3 - Prob. 33ECh. 8.3 - For which square matrices A is there a singular...Ch. 8.3 - Prob. 35ECh. 8.3 - Prob. 36ECh. 8 - The singular values of any diagonal matrix D are...Ch. 8 - Prob. 2ECh. 8 - Prob. 3ECh. 8 - Prob. 4ECh. 8 - Prob. 5ECh. 8 - Prob. 6ECh. 8 - The function q(x1,x2)=3x12+4x1x2+5x2 is a...Ch. 8 - Prob. 8ECh. 8 - If matrix A is positive definite, then all the...Ch. 8 - Prob. 10ECh. 8 - Prob. 11ECh. 8 - Prob. 12ECh. 8 - Prob. 13ECh. 8 - Prob. 14ECh. 8 - Prob. 15ECh. 8 - Prob. 16ECh. 8 - Prob. 17ECh. 8 - Prob. 18ECh. 8 - Prob. 19ECh. 8 - Prob. 20ECh. 8 - Prob. 21ECh. 8 - Prob. 22ECh. 8 - If A and S are invertible nn matrices, then...Ch. 8 - Prob. 24ECh. 8 - Prob. 25ECh. 8 - Prob. 26ECh. 8 - Prob. 27ECh. 8 - Prob. 28ECh. 8 - Prob. 29ECh. 8 - Prob. 30ECh. 8 - Prob. 31ECh. 8 - Prob. 32ECh. 8 - Prob. 33ECh. 8 - Prob. 34ECh. 8 - Prob. 35ECh. 8 - Prob. 36ECh. 8 - Prob. 37ECh. 8 - Prob. 38ECh. 8 - Prob. 39ECh. 8 - Prob. 40ECh. 8 - Prob. 41ECh. 8 - Prob. 42ECh. 8 - Prob. 43ECh. 8 - Prob. 44ECh. 8 - Prob. 45ECh. 8 - Prob. 46ECh. 8 - Prob. 47ECh. 8 - Prob. 48ECh. 8 - Prob. 49ECh. 8 - Prob. 50ECh. 8 - Prob. 51ECh. 8 - Prob. 52ECh. 8 - Prob. 53ECh. 8 - Prob. 54E
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