EBK LINEAR ALGEBRA AND ITS APPLICATIONS
6th Edition
ISBN: 9780135851043
Author: Lay
Publisher: PEARSON CO
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Chapter 2.6, Problem 1PP
Suppose an economy has two sectors: goods and services. One unit of output from goods requires inputs of .2 unit from goods and .5 unit from services. One unit of output from services requires inputs of .4 unit from goods and .3 unit from services. There is a final demand of 20 units of goods and 30 units of services. Set up the Leontief input–output model for tins situation.
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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:
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Subspace W is spanned by the vectors:
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Chapter 2 Solutions
EBK LINEAR ALGEBRA AND ITS APPLICATIONS
Ch. 2.1 - Since vectors in n may be regarded as n 1...Ch. 2.1 - Let A be a 4 4 matrix and let x be a vector in 4....Ch. 2.1 - Suppose A is an m n matrix, all of whose rows are...Ch. 2.1 - If a matrix A is 5 3 and the product AB is 5 7,...Ch. 2.1 - How many rows does B have if BC is a 3 4 matrix?Ch. 2.1 - Let A=[2531] and B=[453k]. What value(s) of k, if...Ch. 2.1 - Let A=[2346], B=[8455], and C=[5231]. Verify that...Ch. 2.1 - Let A=[111123145] and D=[200030005]. Compute AD...Ch. 2.1 - Let A=[3612]. Construct a 2 2 matrix B such that...Ch. 2.1 - Let r1,..., rp be vectors in n, and let Q be an m ...
Ch. 2.1 - Let U be the 3 2 cost matrix described in Example...Ch. 2.1 - If A=[1225] and AB=[121693], determine the first...Ch. 2.1 - Suppose the first two columns, b1 and b2, of B are...Ch. 2.1 - Suppose die third column of B is die sum of die...Ch. 2.1 - Suppose the second column of B is all zeros. What...Ch. 2.1 - Suppose the last column of AB is entirely zero but...Ch. 2.1 - Show that if the columns of B are linearly...Ch. 2.1 - Suppose CA = In (the n n identity matrix). Show...Ch. 2.1 - Suppose AD = Im (the m m identity matrix). Show...Ch. 2.1 - Suppose A is an m n matrix and there exist n m...Ch. 2.1 - Suppose A is a 3 n matrix whose columns span 3....Ch. 2.1 - In Exercises 27 and 28, view vectors in n as n 1...Ch. 2.1 - If u and v are in n. how are uTv and vTu related?...Ch. 2.1 - Prove Theorem 2(b) and 2(c). Use the row-column...Ch. 2.1 - Prove Theorem 2(d). [Hint: The (i, j)-entry in...Ch. 2.1 - Show that ImA = A when A is an m n matrix. You...Ch. 2.1 - Show that AIn = A when A is an m n matrix. [Hint:...Ch. 2.1 - Prove Theorem 3(d). [Hint: Consider the jth row of...Ch. 2.1 - Give a formula for (A Bx)T, where x is a vector...Ch. 2.2 - Use determinants to determine which of the...Ch. 2.2 - Find the inverse of the matrix A = [121156545], if...Ch. 2.2 - If A is an invertible matrix, prove that 5A is an...Ch. 2.2 - Prob. 7ECh. 2.2 - Prob. 8ECh. 2.2 - Let A = [12512], b1 = [13], b2 = [15], b3 = [26],...Ch. 2.2 - Use matrix algebra to show that if A is invertible...Ch. 2.2 - Let A be an invertible n n matrix, and let B be...Ch. 2.2 - Let A be an invertible n n matrix, and let B be...Ch. 2.2 - Suppose AB = AC. where B and C are n p matrices...Ch. 2.2 - Suppose (B C) D = 0, where B and C are m n...Ch. 2.2 - Suppose A, B, and C are invertible n n matrices....Ch. 2.2 - Suppose A and B are n n, B is invertible, and AB...Ch. 2.2 - Solve the equation AB = BC for A, assuming that A,...Ch. 2.2 - Suppose P is invertible and A = PBP1 Solve for B...Ch. 2.2 - If A, B, and C are n n invertible matrices, does...Ch. 2.2 - Suppose A, B, and X are n n matrices with A, X,...Ch. 2.2 - Explain why the columns of an n n; matrix A are...Ch. 2.2 - Explain why the columns of an n n matrix A span n...Ch. 2.2 - Suppose A is n n and the equation Ax = b has a...Ch. 2.2 - Exercises 25 and 26 prove Theorem 4 for A =...Ch. 2.2 - Exercises 25 and 26 prove Theorem 4 for A =...Ch. 2.2 - Exercises 27 and 28 prove special cases of the...Ch. 2.2 - Show that if row 3 of A is replaced by row3(A) 4 ...Ch. 2.2 - Find the inverses of the matrices in Exercises...Ch. 2.2 - Find die inverses of the matrices in Exercises...Ch. 2.2 - Find die inverses of the matrices in Exercises...Ch. 2.2 - Find die inverses of the matrices in Exercises...Ch. 2.2 - Use the algorithm from this section to find the...Ch. 2.2 - Let A = [279256134]. Find the third column of A1...Ch. 2.2 - [M] Let A = [2592754618053715450149]. Find the...Ch. 2.2 - Let A = [121315]. Constuct a 2 3 matrix C (by...Ch. 2.2 - Let A = [11100111]. Construct a 4 2 matrix D...Ch. 2.2 - Let D = [.005.002.001.002.004.002.001.002.005] be...Ch. 2.3 - Determine if A = [234234234] is invertible.Ch. 2.3 - Suppose that for a certain n n matrix A,...Ch. 2.3 - Suppose that A and B are n n matrices and the...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - Unless otherwise specified, assume that all...Ch. 2.3 - An m n upper triangular matrix is one whose...Ch. 2.3 - An m n lower triangular matrix is one whose...Ch. 2.3 - Can a square matrix with two identical columns be...Ch. 2.3 - Is it possible for a 5 5 matrix to be invertible...Ch. 2.3 - If A is invertible, then the columns of A1 are...Ch. 2.3 - If C is 6 6 and the equation Cx = v is consistent...Ch. 2.3 - If the columns of a 7 7 matrix D are linearly...Ch. 2.3 - If n n matrices E and F have the property that EF...Ch. 2.3 - If the equation Gx = y has more than one solution...Ch. 2.3 - If the equation Hx = c is inconsistent for some c...Ch. 2.3 - If an n n matrix K cannot be row reduced to In....Ch. 2.3 - If L is n n and the equation Lx = 0 has the...Ch. 2.3 - Verify the boxed statement preceding Example 1.Ch. 2.3 - Explain why the columns of A2 span n whenever the...Ch. 2.3 - Show that if AB is invertible, so is A. You cannot...Ch. 2.3 - Show that if AB is invertible, so is B.Ch. 2.3 - If A is an n n matrix and the equation Ax = b has...Ch. 2.3 - If A is an n n matrix and the transformation x ...Ch. 2.3 - Suppose A is an n n matrix with the property that...Ch. 2.3 - Suppose A is an n n matrix with the property that...Ch. 2.3 - In Exercises 33 and 34, T is a linear...Ch. 2.3 - In Exercises 33 and 34, T is a linear...Ch. 2.3 - Let T : n n be an invertible linear...Ch. 2.3 - Let T be a linear transformation that maps n onto...Ch. 2.3 - Suppose T and U are linear transformations from n...Ch. 2.3 - Suppose a linear transformation T : n n has the...Ch. 2.3 - Let T : n n be an invertible linear...Ch. 2.3 - Suppose T and S satisfy the invertibility...Ch. 2.4 - Show that[I0AI] is invertible and find its...Ch. 2.4 - Compute XTX, where X is partitioned as [X1 X2].Ch. 2.4 - In Exercises 19, assume that the matrices are...Ch. 2.4 - In Exercises 19, assume that the matrices are...Ch. 2.4 - In Exercises 19, assume that the matrices are...Ch. 2.4 - In Exercises 19, assume that the matrices are...Ch. 2.4 - In Exercises 58, find formulas for X, Y, and Z in...Ch. 2.4 - In Exercises 58, find formulas for X, Y, and Z in...Ch. 2.4 - In Exercises 58, find formulas for X, Y, and Z in...Ch. 2.4 - In Exercises 58, find formulas for X, Y, and Z in...Ch. 2.4 - Suppose A11 is an invertible matrix. Find matrices...Ch. 2.4 - The inverse of [I00CI0ABI] is [I00ZI0XYI]. Find X,...Ch. 2.4 - Let A=[B00C], where B and C are square. Show A is...Ch. 2.4 - Show that the block upper triangular matrix A in...Ch. 2.4 - Suppose A11 is invertible. Find X and Y such that...Ch. 2.4 - Suppose the block matrix A on the left side of (7)...Ch. 2.4 - When a deep space probe is launched, corrections...Ch. 2.4 - a. Verify that A2 = I when A=[1031]. b. Use...Ch. 2.4 - Use partitioned matrices to prove by induction...Ch. 2.4 - Without using row reduction, find the inverse of...Ch. 2.5 - Find an LU factorization of...Ch. 2.5 - In Exercises 16, solve the equation Ax = b by...Ch. 2.5 - In Exercises 16, solve the equation Ax = b by...Ch. 2.5 - In Exercises 16, solve the equation Ax = b by...Ch. 2.5 - In Exercises 16, solve the equation Ax = b by...Ch. 2.5 - In Exercises 16, solve the equation Ax = b by...Ch. 2.5 - In Exercises 16, solve the equation Ax = b by...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - Find an LU factorization of the matrices in...Ch. 2.5 - When A is invertible, MATLAB finds A1 by factoring...Ch. 2.5 - Find A1 as in Exercise 17, using A from Exercise...Ch. 2.5 - Let A be a lower triangular n n matrix with...Ch. 2.5 - Let A = LU be an LU factorization. Explain why A...Ch. 2.5 - Suppose A = BC, where B is invertible. Show that...Ch. 2.5 - (Reduced LU Factorization) With A as in the...Ch. 2.5 - (Rank Factorization) Suppose an m n matrix A...Ch. 2.5 - (QR Factorization) Suppose A = QR, where Q and R...Ch. 2.5 - (Singular Value Decomposition) Suppose A = UDVT,...Ch. 2.5 - (Spectral Factorization) Suppose a 3 3 matrix A...Ch. 2.5 - Design two different ladder networks that each...Ch. 2.5 - Show that if three shunt circuits (with...Ch. 2.5 - Prob. 29ECh. 2.5 - Find a different factorization of the A in...Ch. 2.6 - Suppose an economy has two sectors: goods and...Ch. 2.6 - Exercises 14 refer to an economy that is divided...Ch. 2.6 - Exercises 14 refer to an economy that is divided...Ch. 2.6 - Exercises 14 refer to an economy that is divided...Ch. 2.6 - Exercises 14 refer to an economy that is divided...Ch. 2.6 - Consider the production model x = Cx + d for an...Ch. 2.6 - Repeat Exercise 5 with C=[.1.6.5.2], and d=[1811]....Ch. 2.6 - Let C and d be as in Exercise 5. a. Determine the...Ch. 2.6 - Let C be an n n consumption matrix whose column...Ch. 2.6 - Solve the Leontief production equation for an...Ch. 2.6 - The consumption matrix C for the U.S. economy in...Ch. 2.6 - The Leontief production equation, x = Cx + d, is...Ch. 2.6 - Let C be a consumption matrix such that Cm 0 as m...Ch. 2.7 - Rotation of a figure about a point p in 2 is...Ch. 2.7 - What 3 3 matrix will have the same effect on...Ch. 2.7 - Use matrix multiplication to find the image of the...Ch. 2.7 - In Exercises 38, find the 3 3 matrices that...Ch. 2.7 - In Exercises 38, find the 3 3 matrices that...Ch. 2.7 - In Exercises 38, find the 3 3 matrices that...Ch. 2.7 - In Exercises 38, find the 3 3 matrices that...Ch. 2.7 - In Exercises 38, find the 3 3 matrices that...Ch. 2.7 - In Exercises 38, find the 3 3 matrices that...Ch. 2.7 - A 2 200 data matrix D contains the coordinates of...Ch. 2.7 - Consider the following geometric 2D...Ch. 2.7 - Prob. 11ECh. 2.7 - A rotation in 2 usually requires four...Ch. 2.7 - The usual transformations on homogeneous...Ch. 2.7 - Prob. 14ECh. 2.7 - What vector in 3 has homogeneous coordinates...Ch. 2.7 - Are (1. 2, 3, 4) and (10, 20, 30, 40) homogeneous...Ch. 2.7 - Give the 4 4 matrix that rotates points in 3...Ch. 2.7 - Give the 4 4 matrix that rotates points in 3...Ch. 2.7 - Let S be the triangle with vertices (4.2, 1.2,4),...Ch. 2.7 - Let S be the triangle with vertices (9,3,5),...Ch. 2.7 - [M] The actual color a viewer sees on a screen is...Ch. 2.7 - [M] The signal broadcast by commercial television...Ch. 2.8 - Let A=[115207353] and u=[732] Is u in Nul A? Is u...Ch. 2.8 - Given A=[010001000], find a vector in Nul A and a...Ch. 2.8 - Suppose an n n matrix A is invertible. What can...Ch. 2.8 - Exercises 14 display sets in 2. Assume the sets...Ch. 2.8 - Exercises 14 display sets in 2. Assume the sets...Ch. 2.8 - Exercises 14 display sets in 2. Assume the sets...Ch. 2.8 - Exercises 1-4 display sets in 2. Assume the sets...Ch. 2.8 - Let v1 = [235], v2 = [458], and w = [829]....Ch. 2.8 - Let v1 = [1243], v2 = [4797], v3 = [5865], and u =...Ch. 2.8 - Let v1 = [286], v2 = [387], v3 = [467], p =...Ch. 2.8 - Let v1 = [306], v2 = [223], v3 = [063], and p =...Ch. 2.8 - With A and p as in Exercise 7, determine if p is...Ch. 2.8 - With u = (2, 3, 1) and A as in Exercise 8,...Ch. 2.8 - In Exercises 11 and 12. give integers p and q such...Ch. 2.8 - In Exercises 11 and 12. give integers p and q such...Ch. 2.8 - For A as in Exercise 11, find a nonzero vector in...Ch. 2.8 - For A as in Exercise 12, find a nonzero vector in...Ch. 2.8 - Determine which sets in Exercises 15-20 are bases...Ch. 2.8 - Determine which sets in Exercises 15-20 are bases...Ch. 2.8 - Determine which sets in Exercises 15-20 are bases...Ch. 2.8 - Determine which sets in Exercises 15-20 are bases...Ch. 2.8 - Determine which sets in Exercises 15-20 are bases...Ch. 2.8 - Determine which sets in Exercises 15-20 are bases...Ch. 2.8 - Exercises 23-26 display a matrix A and an echelon...Ch. 2.8 - Exercises 23-26 display a matrix A and an echelon...Ch. 2.8 - Exercises 23-26 display a matrix A and an echelon...Ch. 2.8 - Exercises 23-26 display a matrix A and an echelon...Ch. 2.8 - Construct a nonzero 3 3 matrix A and a nonzero...Ch. 2.8 - Construct a nonzero 3 3 matrix A and a vector b...Ch. 2.8 - Construct a nonzero 3 3 matrix A and a nonzero...Ch. 2.8 - Suppose the columns of a matrix A = [a1 ap] are...Ch. 2.8 - In Exercises 31-36, respond as comprehensively as...Ch. 2.8 - In Exercises 31-36. respond as comprehensively as...Ch. 2.8 - In Exercises 31-36, respond as comprehensively as...Ch. 2.8 - In Exercises 31-36, respond as comprehensively as...Ch. 2.8 - In Exercises 31-36, respond as comprehensively as...Ch. 2.8 - In Exercises 31-36, respond as comprehensively as...Ch. 2.8 - [M] In Exercises 37 and 38, construct bases for...Ch. 2.8 - [M] In Exercises 37 and 38, construct bases for...Ch. 2.9 - Determine the dimension of the subspace H of 3...Ch. 2.9 - Prob. 2PPCh. 2.9 - Could 3 possibly contain a four-dimensional...Ch. 2.9 - In Exercises 1 and 2, find the vector x determined...Ch. 2.9 - In Exercises 1 and 2, find the vector x determined...Ch. 2.9 - In Exercises 3-6, the vector s is in a subspace H...Ch. 2.9 - In Exercises 1 and 2, find the vector x determined...Ch. 2.9 - In Exercises 3-6, the vector x is in a subspace H...Ch. 2.9 - In Exercises 3-6, the vector x is in a subspace H...Ch. 2.9 - Let b1 = [30], b2 = [12], w = [72], x = [41], and...Ch. 2.9 - Let b1 = [02], b2 = [21], x = [23], y = [24], z =...Ch. 2.9 - Exercises 9-12 display a matrix A and an echelon...Ch. 2.9 - Exercises 9-12 display a matrix A and an echelon...Ch. 2.9 - Exercises 9-12 display a matrix A and an echelon...Ch. 2.9 - Exercises 9-12 display a matrix A and an echelon...Ch. 2.9 - In Exercises 13 and 14, find a basis for the...Ch. 2.9 - In Exercises 13 and 14, find a basis for the...Ch. 2.9 - Suppose a 3 5 matrix A has three pivot columns....Ch. 2.9 - Suppose a 4 7 matrix A has three pivot columns....Ch. 2.9 - If the subspace of all solutions of Ax = 0 has a...Ch. 2.9 - What is the rank of a 4 5 matrix whose null space...Ch. 2.9 - If the tank of a 7 6 matrix A is 4, what is the...Ch. 2.9 - Show that a set of vectors {v1, v2, , v5} in n is...Ch. 2.9 - If possible, construct a 3 4 matrix A such that...Ch. 2.9 - Constructa4 3 matrix with tank 1.Ch. 2.9 - Let A be an n p matrix whose column space is...Ch. 2.9 - Suppose columns 1, 3, 5, and 6 of a matrix A are...Ch. 2.9 - Suppose vectors b1, bp span a subspace W, and let...Ch. 2.9 - Prob. 37ECh. 2.9 - [M] Let H = Span {v1, v2, v3} and B= {v1, v2,...Ch. 2 - Find the matrix C whose inverse is C1 = [4567].Ch. 2 - Show that A = [000100010]. Show that A3 = 0. Use...Ch. 2 - Suppose An = 0 for some n 1. Find an inverse for...Ch. 2 - Suppose an n n matrix A satisfies the equation A2...Ch. 2 - Prob. 20SECh. 2 - Let A = [1382411125] and B = [351534]. Compute A1B...Ch. 2 - Find a matrix A such that the transformation x Ax...Ch. 2 - Suppose AB =[5423] and B = [7321]. Find A.Ch. 2 - Suppose A is invertible. Explain why ATA is also...Ch. 2 - Let x1, , xn, be fixed numbers. The matrix below,...Ch. 2 - Prob. 26SECh. 2 - Given u in n with uTu = 1, Let P = uuT (an outer...Ch. 2 - Prob. 28SECh. 2 - Prob. 29SECh. 2 - Let A be an n n singular matrix Describe how to...Ch. 2 - Let A be a 6 4 matrix and B a 4 6 matrix. Show...Ch. 2 - Suppose A is a 5 3 matrix and mere exists a 3 5...Ch. 2 - Prob. 33SECh. 2 - [M] Let An be the n n matrix with 0s on the main...
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