Linear Algebra with Applications (2-Download)
5th Edition
ISBN: 9780321796974
Author: Otto Bretscher
Publisher: PEARSON
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Textbook Question
Chapter 3.3, Problem 29E
Find a basis of the subspace of
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Find a basis for the space of all 2 by 3 matrices whose columns add to zero. Find a basis for the subspace whose rows also add to zero.
Chapter 3 Solutions
Linear Algebra with Applications (2-Download)
Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...
Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 1 through 13, find...Ch. 3.1 - For each matrix A in Exercises 14 through 16, find...Ch. 3.1 - For each matrix A in Exercises 14 through 16, find...Ch. 3.1 - For each matrix A in Exercises 14 through 16, find...Ch. 3.1 - For each matrix A in Exercises 17 through 22,...Ch. 3.1 - For each matrix A in Exercises 17 through 22,...Ch. 3.1 - For each matrix A in Exercises 17 through 22,...Ch. 3.1 - For each matrix A in Exercises 17 through 22,...Ch. 3.1 - For each matrix A in Exercises 17 through 22,...Ch. 3.1 - For each matrix A in Exercises 17 through 22,...Ch. 3.1 - Describe the images and kernels of the...Ch. 3.1 - Prob. 24ECh. 3.1 - Describe the images and kernels of the...Ch. 3.1 - What is the image of a function f from to given...Ch. 3.1 - Give an example of a noninvertible function f from...Ch. 3.1 - Prob. 28ECh. 3.1 - Give an example of a function whose image is the...Ch. 3.1 - Give an example of a matrix A such that im(A)...Ch. 3.1 - Give an example of a matrix A such that im(A) is...Ch. 3.1 - Give an example of a linear transformation whose...Ch. 3.1 - Give an example of a linear transformation whose...Ch. 3.1 - Give an example of a linear transformation whose...Ch. 3.1 - Consider a nonzero vector v in 3 . Arguing...Ch. 3.1 - Prob. 36ECh. 3.1 - For the matrix A=[010001000] , describe the images...Ch. 3.1 - Consider a square matrix A. a. What is the...Ch. 3.1 - Consider an np matrix A and a pm matrix B. a. What...Ch. 3.1 - Consider an np matrix A and a pm matrix B. If...Ch. 3.1 - Consider the matrix A=[0.360.480.480.64] . a....Ch. 3.1 - Express the image of the matrix...Ch. 3.1 - Prob. 43ECh. 3.1 - Consider a matrix A, and let B=rref(A) . a. Is...Ch. 3.1 - Prob. 45ECh. 3.1 - Prob. 46ECh. 3.1 - Prob. 47ECh. 3.1 - Consider a 22 matrix A with A2=A . a. If w is in...Ch. 3.1 - Verify that the kernel of a linear transformation...Ch. 3.1 - Consider a square matrix A with ker(A2)=ker(A3) ....Ch. 3.1 - Consider an np matrix A and a pm in matrix B...Ch. 3.1 - Prob. 52ECh. 3.1 - In Exercises 53 and 54, we will work with the...Ch. 3.1 - See Exercise 53 for some background. When...Ch. 3.2 - Which of the sets W in Exercises 1 through 3 are...Ch. 3.2 - Which of the sets W in Exercises 1 through 3 are...Ch. 3.2 - Which of the sets W in Exercises 1 through 3 are...Ch. 3.2 - Consider the vectors v1,v2,...,vm in n . Is span...Ch. 3.2 - Give a geometrical description of all subspaces of...Ch. 3.2 - Consider two subspaces V and W of n . a. Is the...Ch. 3.2 - Consider a nonempty subset W of n that is closed...Ch. 3.2 - Find a nontrivial relation among the following...Ch. 3.2 - Consider the vectors v1,v2,...,vm in n , with vm=0...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 10 through 20, use paper and pencil...Ch. 3.2 - In Exercises 21 through 26, find a redundant...Ch. 3.2 - In Exercises 21 through 26, find a redundant...Ch. 3.2 - In Exercises 21 through 26, find a redundant...Ch. 3.2 - In Exercises 21 through 26, find a redundant...Ch. 3.2 - In Exercises 21 through 26, find a redundant...Ch. 3.2 - In Exercises 21 through 26, find a redundant...Ch. 3.2 - Find a basis of the image of the matrices in...Ch. 3.2 - Find a basis of the image of the matrices in...Ch. 3.2 - Find a basis of the image of the matrices in...Ch. 3.2 - Find a basis of the image of the matrices in...Ch. 3.2 - Find a basis of the image of the matrices in...Ch. 3.2 - Find a basis of the image of the matrices in...Ch. 3.2 - Prob. 33ECh. 3.2 - Consider the 54 matrix A=[ v 1 v 2 v 3 v 4] ....Ch. 3.2 - Prob. 35ECh. 3.2 - Consider a linear transformation T from n to p...Ch. 3.2 - Consider a linear transformation T from n to p...Ch. 3.2 - Prob. 38ECh. 3.2 - Consider some linearly independent vectors...Ch. 3.2 - Consider an np matrix A and a pm matrix B. Weare...Ch. 3.2 - Prob. 41ECh. 3.2 - Consider some perpendicular unit vectors...Ch. 3.2 - Consider three linearly independent vectors...Ch. 3.2 - Consider linearly independent vectors v1,v2,...,vm...Ch. 3.2 - Prob. 45ECh. 3.2 - Find a basis of the kernel of the matrix...Ch. 3.2 - Consider three linearly independent vectors...Ch. 3.2 - Express the plane V in 3 with equation...Ch. 3.2 - Express the line L in 3 spanned by the vector...Ch. 3.2 - Consider two subspaces V and W of n . Let V+W...Ch. 3.2 - Prob. 51ECh. 3.2 - Prob. 52ECh. 3.2 - Consider a subspace V of n . We define the...Ch. 3.2 - Consider the line L spanned by [123] in 3 . Find a...Ch. 3.2 - Consider the subspace L of 5 spanned by the...Ch. 3.2 - Prob. 56ECh. 3.2 - Consider the matrix...Ch. 3.2 - Prob. 58ECh. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 1 through 20, find the redundant...Ch. 3.3 - In Exercises 21 through 25, find the reduced...Ch. 3.3 - In Exercises 21 through 25, find the reduced...Ch. 3.3 - In Exercises 21 through 25, find the reduced...Ch. 3.3 - In Exercises 21 through 25, find the reduced...Ch. 3.3 - In Exercises 21 through 25, find the reduced...Ch. 3.3 - Consider the matrices C=[ 1 1 1 1 0 0 1 1 1],H=[ 1...Ch. 3.3 - Determine whether the following vectors form a...Ch. 3.3 - For which value(s) of the constant k do the...Ch. 3.3 - Find a basis of the subspace of 3 defined by...Ch. 3.3 - Find a basis of the subspace of 4 defined by the...Ch. 3.3 - Let V be the subspace of 4 defined by the equation...Ch. 3.3 - Find a basis of the subspace of 4 that consists of...Ch. 3.3 - A subspace V of n is called a hyperplane if V...Ch. 3.3 - Consider a subspace V in m that is defined by...Ch. 3.3 - Consider a nonzero vector v in n . What is the...Ch. 3.3 - Can you find a 33 matrix A such that im(A)=ker(A)...Ch. 3.3 - Give an example of a 45 matrix A with dim(kerA)=3...Ch. 3.3 - a. Consider a linear transformation T from 5 to 3...Ch. 3.3 - Prob. 39ECh. 3.3 - Prob. 40ECh. 3.3 - Prob. 41ECh. 3.3 - In Exercises 40 through 43, consider the problem...Ch. 3.3 - Prob. 43ECh. 3.3 - For Exercises 44 through 61, consider the problem...Ch. 3.3 - Prob. 45ECh. 3.3 - Prob. 46ECh. 3.3 - Prob. 47ECh. 3.3 - Prob. 48ECh. 3.3 - Prob. 49ECh. 3.3 - Prob. 50ECh. 3.3 - Prob. 51ECh. 3.3 - Prob. 52ECh. 3.3 - Prob. 53ECh. 3.3 - For Exercises 44 through 61, consider the problem...Ch. 3.3 - Prob. 55ECh. 3.3 - Prob. 56ECh. 3.3 - Prob. 57ECh. 3.3 - Prob. 58ECh. 3.3 - Prob. 59ECh. 3.3 - Prob. 60ECh. 3.3 - Find all points P in the plane such that you can...Ch. 3.3 - Prob. 62ECh. 3.3 - Consider two subspaces V and W of n , where Vis...Ch. 3.3 - Consider a subspace V of n with dim(V)=n . Explain...Ch. 3.3 - Consider two subspaces V and W of n , with VW={0}...Ch. 3.3 - Two subspaces V and W of n arc called...Ch. 3.3 - Consider linearly independent vectors v1,v2,...vp...Ch. 3.3 - Use Exercise 67 to construct a basis of 4 that...Ch. 3.3 - Consider two subspaces V and W of n . Show that...Ch. 3.3 - Use Exercise 69 to answer the following question:...Ch. 3.3 - Prob. 71ECh. 3.3 - Prob. 72ECh. 3.3 - Prob. 73ECh. 3.3 - Prob. 74ECh. 3.3 - Prob. 75ECh. 3.3 - Consider the matrix A=[1221] . Find scalars...Ch. 3.3 - Prob. 77ECh. 3.3 - An nn matrix A is called nilpotent if Am=0 for...Ch. 3.3 - Consider a nilpotent nn matrix A. Use the...Ch. 3.3 - Prob. 80ECh. 3.3 - Prob. 81ECh. 3.3 - If a 33 matrix A represents the projection onto a...Ch. 3.3 - Consider a 42 matrix A and a 25 matrix B. a. What...Ch. 3.3 - Prob. 84ECh. 3.3 - Prob. 85ECh. 3.3 - Prob. 86ECh. 3.3 - Prob. 87ECh. 3.3 - Prob. 88ECh. 3.3 - Prob. 89ECh. 3.3 - Prob. 90ECh. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 1 through 18, determine whether the...Ch. 3.4 - In Exercises 19 through 24, find the matrix B of...Ch. 3.4 - In Exercises 19 through 24, find the matrix B of...Ch. 3.4 - In Exercises 19 through 24, find the matrix B of...Ch. 3.4 - In Exercises 19 through 24, find the matrix B of...Ch. 3.4 - In Exercises 19 through 24, find the matrix B of...Ch. 3.4 - In Exercises 19 through 24, find the matrix B of...Ch. 3.4 - In Exercises 25 through 30, find the matrix B of...Ch. 3.4 - In Exercises 25 through 30, find the matrix B of...Ch. 3.4 - In Exercises 25 through 30, find the matrix B of...Ch. 3.4 - In Exercises 25 through 30, find the matrix B of...Ch. 3.4 - In Exercises 25 through 30, find the matrix B of...Ch. 3.4 - In Exercises 25 through 30, find the matrix B of...Ch. 3.4 - Let =(v1,v2,v3)be any basis of 3consisting of...Ch. 3.4 - Let =(v1,v2,v3)be any basis of 3consisting of...Ch. 3.4 - Let =(v1,v2,v3)be any basis of 3consisting of...Ch. 3.4 - Let =(v1,v2,v3)be any basis of 3consisting of...Ch. 3.4 - Let =(v1,v2,v3)be any basis of 3consisting of...Ch. 3.4 - Let =(v1,v2,v3)be any basis of 3consisting of...Ch. 3.4 - In Exercises 37 through 42, find a basis of n such...Ch. 3.4 - In Exercises 37 through 42, find a basis of n such...Ch. 3.4 - In Exercises 37 through 42, find a basis of n such...Ch. 3.4 - Prob. 40ECh. 3.4 - In Exercises 37 through 42, find a basis of n such...Ch. 3.4 - In Exercises 37 through 42, find a basis of n such...Ch. 3.4 - Consider the plane x1+2x2+x3=0 with basis...Ch. 3.4 - Consider the plane 2x13x2+4x3=0 with basis...Ch. 3.4 - Consider the plane 2x13x2+4x3=0. Find a basis of...Ch. 3.4 - Consider the plane x1+2x2+x3=0. Find a basis of...Ch. 3.4 - Consider a linear transformation T from 2 to 2...Ch. 3.4 - In the accompanying figure, sketch the vector x...Ch. 3.4 - Prob. 49ECh. 3.4 - Given a hexagonal tiling of the plane, such as you...Ch. 3.4 - Prob. 51ECh. 3.4 - If is a basis of n , is the transformation T from...Ch. 3.4 - Consider the basis of 2 consisting of the vectors...Ch. 3.4 - Let be the basis of n consisting of the vectors...Ch. 3.4 - Consider the basis of 2 consisting of the vectors...Ch. 3.4 - Find a basis of 2 such that andCh. 3.4 - Show that if a 33 matrix A represents the...Ch. 3.4 - Consider a 33 matrix A and a vector v in 3...Ch. 3.4 - Is matrix [2003] similar to matrix [2113] ?Ch. 3.4 - Is matrix [1001] similar to matrix [0110] ?Ch. 3.4 - Find a basis of 2 such that the matrix of the...Ch. 3.4 - Find a basis of 2 such that the matrix of the...Ch. 3.4 - Prob. 63ECh. 3.4 - Is matrix [abcd] similar to matrix [acbd] for all...Ch. 3.4 - Prove parts (a) and (b) of Theorem 3.4.6.Ch. 3.4 - Consider a matrix A of the form A=[abba] , where...Ch. 3.4 - If c0 ,find the matrix of the linear...Ch. 3.4 - Prob. 68ECh. 3.4 - If A is a 22 matrix such that A[12]=[36] and...Ch. 3.4 - Is there a basis of 2 such that matrix B of...Ch. 3.4 - Suppose that matrix A is similar to B, with B=S1AS...Ch. 3.4 - If A is similar to B, what is the relationship...Ch. 3.4 - Prob. 73ECh. 3.4 - Consider the regular tetrahedron in the...Ch. 3.4 - Prob. 75ECh. 3.4 - Prob. 76ECh. 3.4 - Prob. 77ECh. 3.4 - This problem refers to Leontief’s input—output...Ch. 3.4 - Prob. 79ECh. 3.4 - Prob. 80ECh. 3.4 - Consider the linear transformation...Ch. 3.4 - Prob. 82ECh. 3 - If v1,v2,...,vn and w1,w2,...,wm are any twobases...Ch. 3 - If A is a 56 matrix of rank 4, then the nullity of...Ch. 3 - The image of a 34 matrix is a subspace of 4 .Ch. 3 - The span of vectors v1,v2,...,vn consists of all...Ch. 3 - Prob. 5ECh. 3 - Prob. 6ECh. 3 - The kernel of any invertible matrix consists of...Ch. 3 - The identity matrix In is similar to all...Ch. 3 - Prob. 9ECh. 3 - The column vectors of a 54 matrix must be...Ch. 3 - Prob. 11ECh. 3 - Prob. 12ECh. 3 - Prob. 13ECh. 3 - Prob. 14ECh. 3 - Prob. 15ECh. 3 - Vectors [100],[210],[321] form a basis of 3 .Ch. 3 - Prob. 17ECh. 3 - Prob. 18ECh. 3 - Prob. 19ECh. 3 - Prob. 20ECh. 3 - Prob. 21ECh. 3 - Prob. 22ECh. 3 - Prob. 23ECh. 3 - Prob. 24ECh. 3 - Prob. 25ECh. 3 - If a 22 matrix P represents the orthogonal...Ch. 3 - Prob. 27ECh. 3 - Prob. 28ECh. 3 - Prob. 29ECh. 3 - Prob. 30ECh. 3 - Prob. 31ECh. 3 - Prob. 32ECh. 3 - Prob. 33ECh. 3 - Prob. 34ECh. 3 - Prob. 35ECh. 3 - If A and B are nn matrices, and vector v is in...Ch. 3 - Prob. 37ECh. 3 - Prob. 38ECh. 3 - Prob. 39ECh. 3 - Prob. 40ECh. 3 - Prob. 41ECh. 3 - If two nn matrices A and B have the same rank,...Ch. 3 - Prob. 43ECh. 3 - If A2=0 for a 1010 matrix A, then the inequality...Ch. 3 - Prob. 45ECh. 3 - Prob. 46ECh. 3 - Prob. 47ECh. 3 - Prob. 48ECh. 3 - Prob. 49ECh. 3 - Prob. 50ECh. 3 - Prob. 51ECh. 3 - Prob. 52ECh. 3 - Prob. 53E
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- Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.arrow_forwardProve that if A is similar to B and A is diagonalizable, then B is diagonalizable.arrow_forwardIn Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[aba]}arrow_forward
- In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[aba+b+1]}arrow_forwardIn Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[a0a]}arrow_forwardFind a basis for the subspace of R4 determined by the equation 2x-3y+4z-w=0arrow_forward
- Find a basis of the subspace of R* consisting of all vectors of the form x1 -9x1+ x2 -9x1– 2x2 -6x1+4x2] Your answer should be a list of row vectors separated by commas. (Click vector to learn about entering vectors.) Basis = {arrow_forwardFind a basis of the subspace of IR³ spanned by the following vectors: 3 -3 3 -3 -3 6 -9 Answer: 4 2 -6 10 -2 -4 5 -8 3 -3 -2 6 -9 3 -3 -2 6 -9arrow_forwardLet W be the subspace of the vector space of polynomials of degree 3 spanned by 2₁ = 1 + 3x - 2x² + x³, v₂ = -3-9x + 6x²-3x³, v3 = x + x². x+x² - 2x³, 24 = 1+ 2x - 3x² + 3x³, vs = -2-5x + 5x2 - 4x3. Which of the following statements are true? (i) The set Bw = {₁, 23,1₁) is a basis for W. (ii) The set Cw = {₁, 4) is a basis for W. (iii) The set By = {1, 23, 1, 2} is a basis for the space of polynomials of degree 3. (iv) The set Cy (v1,vs, 1,22} is a basis for the space of polynomials of (v) a +bx+cx² + da3 is in W, if 5a - b + d = 0 and 7a+2b+c=0 (vi) 1-x+x²-23 EW and -1-4x+x² + x³ € W. degree 3. A) () and (v) B) (i), (iii) and (iv) C) (i), (ii), and (vi) OD) (ii) and (iv) E) (), (iv) and (v)arrow_forward
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