Linear Algebra with Applications (2-Download)
5th Edition
ISBN: 9780321796974
Author: Otto Bretscher
Publisher: PEARSON
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Chapter 3.3, Problem 30E
Find a basis of the subspace of
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You are given a plane Π in R3 defined by two vectors, p1 and p2, and a subspace W in R3 spanned by twovectors, w1 and w2. Your task is to project the plane Π onto the subspace W.First, answer the question of what the projection matrix is that projects onto the subspace W and how toapply it to find the desired projection. Second, approach the task in a different way by using the Gram-Schmidtmethod to find an orthonormal basis for subspace W, before then using the resulting basis vectors for theprojection. Last, compare the results obtained from both methods
Plane II is spanned by the vectors:
- (2) · P² - (4)
P1=2
P21
3
Subspace W is spanned by the vectors:
2
W1
- (9) ·
1
W2
1
= (³)
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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