Linear Algebra and Its Applications (5th Edition)
5th Edition
ISBN: 9780321982384
Author: David C. Lay, Steven R. Lay, Judi J. McDonald
Publisher: PEARSON
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Textbook Question
Chapter 1.5, Problem 30E
In Exercises 29-32, (a) does the equation Ax = 0 have a nontrivial solution and (b) does the equation Ax = b have at least one solution for every possible b?
30. A is a 3 × 3 matrix with two pivot positions.
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Într-un bloc sunt apartamente cu 2 camere și apartamente cu 3 camere , în total 20 de apartamente și 45 de camere.Calculați câte apartamente sunt cu 2 camere și câte apartamente sunt cu 3 camere.
1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set
Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k
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Question 3
over a field K.
In this question, MË(K) denotes the set of n × n matrices
(a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is
equivalent to A-¹? Justify your answer.
(b) Let B be given by
8
B = 0 7 7
0 -7 7
Working over the field F2 with 2 elements, compute the rank of B as an element
of M2(F2).
(c) Let
1
C
-1 1
[4]
[6]
and consider C as an element of M3(Q). Determine the minimal polynomial
mc(x) and hence, or otherwise, show that C can not be diagonalised.
[7]
(d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write
down all the eigenvalues. Show your working.
[8]
Chapter 1 Solutions
Linear Algebra and Its Applications (5th Edition)
Ch. 1.1 - State in words the next elementary row operation...Ch. 1.1 - The augmented matrix of a linear system has been...Ch. 1.1 - Is (3, 4, 2) a solution of the following system?...Ch. 1.1 - For what values of h and k is the following system...Ch. 1.1 - Solve each system in Exercises 1-4 by using...Ch. 1.1 - Solve each system in Exercises 1-4 by using...Ch. 1.1 - Find the point (x1, x2) that lies on the line x1 +...Ch. 1.1 - Find the point of intersection of the lines x1 ...Ch. 1.1 - Consider each matrix in Exercises 5 and 6 as the...Ch. 1.1 - Consider each matrix in Exercises 5 and 6 as the...
Ch. 1.1 - In Exercises 7-10, the augmented matrix of a...Ch. 1.1 - In Exercises 7-10, the augmented matrix of a...Ch. 1.1 - In Exercises 7-10, the augmented matrix of a...Ch. 1.1 - In Exercises 7-10, the augmented matrix of a...Ch. 1.1 - Solve the systems in Exercises 11-14. 11....Ch. 1.1 - Solve the systems in Exercises 11-14. 12....Ch. 1.1 - Solve the systems in Exercises 11-14. 13....Ch. 1.1 - Solve the systems in Exercises 11-14....Ch. 1.1 - Determine if the systems in Exercises 15 and 16...Ch. 1.1 - Determine if the systems in Exercises 15 and 16...Ch. 1.1 - Do the three lines x1 4x2 = 1, 2x1 x2 = 3, and...Ch. 1.1 - Do the three planes x1 + 2x2 + x3 = 4, x2 x3 = 1,...Ch. 1.1 - In Exercises 19-22, determine the value(s) of h...Ch. 1.1 - In Exercises 19-22, determine the value(s) of h...Ch. 1.1 - In Exercises 19-22, determine the value(s) of h...Ch. 1.1 - In Exercises 19-22, determine the value(s) of h...Ch. 1.1 - In Exercises 23 and 24, key statements from this...Ch. 1.1 - In Exercises 23 and 24, key statements from this...Ch. 1.1 - Find an equation involving g, h, and k that makes...Ch. 1.1 - Construct three different augmented matrices for...Ch. 1.1 - Suppose the system below is consistent for all...Ch. 1.1 - Suppose a, b, c, and d are constants such that a...Ch. 1.1 - In Exercises 29-32, find the elementary row...Ch. 1.1 - In Exercises 29-32, find the elementary row...Ch. 1.1 - In Exercises 29-32, find the elementary row...Ch. 1.1 - In Exercises 29-32, find the elementary row...Ch. 1.1 - An important concern in the study of heat transfer...Ch. 1.1 - Solve the system of equations from Exercise 33....Ch. 1.2 - Find the general solution of the linear system...Ch. 1.2 - Find the general solution of the system...Ch. 1.2 - Suppose a 4 7 coefficient matrix for a system of...Ch. 1.2 - In Exercises 1 and 2, determine which matrices arc...Ch. 1.2 - In Exercises 1 and 2, determine which matrices are...Ch. 1.2 - Row reduce the matrices in Exercises 3 and 4 to...Ch. 1.2 - Row reduce the matrices in Exercises 3 and 4 to...Ch. 1.2 - Describe the possible echelon forms of a nonzero 2...Ch. 1.2 - Repeat Exercise 5 for a nonzero 3 2 matrix. 5....Ch. 1.2 - Find the general solutions of the systems whose...Ch. 1.2 - Find the general solutions of the systems whose...Ch. 1.2 - Find the general solutions of the systems whose...Ch. 1.2 - Find the general solutions of the systems whose...Ch. 1.2 - Find the general solutions of the systems whose...Ch. 1.2 - Find the general solutions of the systems whose...Ch. 1.2 - Find the general solutions of the systems whose...Ch. 1.2 - Find the general solutions of the systems whose...Ch. 1.2 - Exercises 15 and 16 use the notation of Example 1...Ch. 1.2 - Exercises 15 and 16 use the notation of Example 1...Ch. 1.2 - In Exercises 17 and 18, determine the value(s) of...Ch. 1.2 - In Exercises 17 and 18, determine the value(s) of...Ch. 1.2 - In Exercises 19 and 20, choose h and k such that...Ch. 1.2 - In Exercises 19 and 20, choose h and k such that...Ch. 1.2 - In Exercises 21 and 22, mark each statement True...Ch. 1.2 - In Exercises 21 and 22, mark each statement True...Ch. 1.2 - Suppose a 3 5 coefficient matrix for a system has...Ch. 1.2 - Suppose a system of linear equations has a 3 5...Ch. 1.2 - Suppose the coefficient matrix of a system of...Ch. 1.2 - Suppose the coefficient matrix of a linear system...Ch. 1.2 - Restate the last sentence in Theorem 2 using the...Ch. 1.2 - What would you have to know about the pivot...Ch. 1.2 - A system of linear equations with fewer equations...Ch. 1.2 - Give an example of an inconsistent underdetermined...Ch. 1.2 - A system of linear equations with more equations...Ch. 1.2 - Suppose an n (n + 1) matrix is row reduced to...Ch. 1.2 - Find the interpolating polynomial p(t) = a0 + a1t...Ch. 1.2 - [M] In a wind tunnel experiment, die force on a...Ch. 1.3 - Prob. 1PPCh. 1.3 - For what value(s) of h will y be in Span{v1, v2,...Ch. 1.3 - Let w1, w2, w3, u, and v be vectors in n. Suppose...Ch. 1.3 - In Exercises 1 and 2, compute u + v and u 2v. 1....Ch. 1.3 - In Exercises 1 and 2, compute u + v and u 2v. 2....Ch. 1.3 - In Exercises 3 and 4, display the following...Ch. 1.3 - In Exercises 3 and 4, display the following...Ch. 1.3 - In Exercises 5 and 6, write a system of equations...Ch. 1.3 - In Exercises 5 and 6, write a system of equations...Ch. 1.3 - Use the accompanying figure to write each vector...Ch. 1.3 - Use the accompanying figure to write each vector...Ch. 1.3 - In Exercises 9 and 10, write a vector equation...Ch. 1.3 - In Exercises 9 and 10, write a vector equation...Ch. 1.3 - In Exercises 11 and 12, determine if b is a linear...Ch. 1.3 - In Exercises 11 and 12, determine if b is a linear...Ch. 1.3 - In Exercises 13 and 14, determine if b is a linear...Ch. 1.3 - In Exercises 13 and 14, determine if b is a linear...Ch. 1.3 - In Exercises 15 and 16, list five vectors in Span...Ch. 1.3 - In Exercises 15 and 16, list five vectors in Span...Ch. 1.3 - Let a1=[142],a2=[237],andb=[41h]. For what...Ch. 1.3 - Let v1=[102],v2=[318],andy=[h53]. For what...Ch. 1.3 - Give a geometric description of Span {v1, v2} for...Ch. 1.3 - Give a geometric description of Span {v1, v2} for...Ch. 1.3 - Let u=[21]andv=[21]. Show that [hk] is an Span {u,...Ch. 1.3 - Construct a 3 3 matrix A, with nonzero entries,...Ch. 1.3 - a. Another notation for the vector [43] is [-4 3]....Ch. 1.3 - a. Any list of five real numbers is a vector in 5....Ch. 1.3 - Let A = [104032263] and b = [414]. Denote the...Ch. 1.3 - Let A = [206185121], let b = [1033], let W be the...Ch. 1.3 - A mining company has two mines. One days operation...Ch. 1.3 - A steam plain bums two types of coal: anthracite...Ch. 1.3 - Let v1, vk be points in 3 and suppose that for j...Ch. 1.3 - Let v be the center of mass of a system of point...Ch. 1.3 - A thin triangular plate of uniform density and...Ch. 1.3 - Consider the vectors v1, v2, v3, and b in 2, shown...Ch. 1.3 - Use the vectors u = (u1, , un), v = (v1, , vn),...Ch. 1.3 - Use the vector u = (u1, , un) to verify the...Ch. 1.4 - Let A = [152031954817], P = [3204], and b = [790]....Ch. 1.4 - Let A = [2531], u = [41], and v = [35]. Verify...Ch. 1.4 - Construct a 3 3 matrix A and vectors b and c in 3...Ch. 1.4 - Compute the products in Exercises 1-4 using (a)...Ch. 1.4 - Compute the products in Exercises 1-4 using (a)...Ch. 1.4 - Compute the products in Exercises 1-4 using (a)...Ch. 1.4 - Compute the products in Exercises 1-4 using (a)...Ch. 1.4 - In Exercises 5-8, use the definition of Ax to...Ch. 1.4 - In Exercises 5-8, use the definition of Ax to...Ch. 1.4 - In Exercises 5-8, use the definition of Ax to...Ch. 1.4 - In Exercises 5-8, use the definition of Ax to...Ch. 1.4 - In Exercises 9 and 10, write the system first as a...Ch. 1.4 - In Exercises 9 and 10, write the system first as a...Ch. 1.4 - Given A and b in Exercises 11 and 12, write the...Ch. 1.4 - Given A and b in Exercises 11 and 12, write the...Ch. 1.4 - Let u = [044] and A = [352611]. Is u in the plane...Ch. 1.4 - Let u = [232] and A = [587011130]. Is u in the...Ch. 1.4 - Let A = [2163] and b = [b1b2]. Show that the...Ch. 1.4 - Repeat Exercise 15: A = [134326518], b = [b1b2b3]....Ch. 1.4 - Exercises 17-20 refer to the matrices A and B...Ch. 1.4 - Exercises 17-20 refer to the matrices A and B...Ch. 1.4 - Exercises 17-20 refer to the matrices A and B...Ch. 1.4 - Exercises 17-20 refer to the matrices A and B...Ch. 1.4 - Let v1 = [1010], v2 = [0101], v3 = [1001]. Does...Ch. 1.4 - Let v1 = [002], v2 = [038], v3 = [415]. Does {v1,...Ch. 1.4 - a. The equation Ax = b is referred to as a vector...Ch. 1.4 - a. Every matrix equation Ax = b corresponds to a...Ch. 1.4 - Note that [431525623][312]=[7310]. Use this fact...Ch. 1.4 - Let u = [725], v = [313], and w = [610]. It can be...Ch. 1.4 - Let q1, q2, q3, and v represent vectors in 5, and...Ch. 1.4 - Rewrite the (numerical) matrix equation below in...Ch. 1.4 - Construct a 3 3 matrix, not in echelon form,...Ch. 1.4 - Construct a 3 3 matrix, not in echelon form,...Ch. 1.4 - Let A be a 3 2 matrix. Explain why the equation...Ch. 1.4 - Could a set of three vectors in 4 span all of 4?...Ch. 1.4 - Suppose A is a 4 3 matrix and b is a vector in 4...Ch. 1.4 - Suppose A is a 3 3 matrix and b is a vector in 3...Ch. 1.4 - Let A be a 3 4 matrix, let y1 and y2 be vectors...Ch. 1.4 - Let A be a 5 3 matrix, let y be a vector in 3,...Ch. 1.4 - [M] In Exercises 37-40, determine if the columns...Ch. 1.4 - [M] In Exercises 37-40, determine if the columns...Ch. 1.4 - [M] In Exercises 37-40, determine if the columns...Ch. 1.4 - [M] In Exercises 37-40, determine if the columns...Ch. 1.5 - Each of the following equations determines a plane...Ch. 1.5 - Write the general solution of 10x1 3x2 2x3 = 7...Ch. 1.5 - Prove the first pan of Theorem 6: Suppose that p...Ch. 1.5 - In Exercises 1-4, determine if the system has a...Ch. 1.5 - In Exercises 1-4, determine if the system has a...Ch. 1.5 - In Exercises 1-4, determine if the system has a...Ch. 1.5 - In Exercises 1-4, determine if the system has a...Ch. 1.5 - In Exercises 5 and 6, follow the method of...Ch. 1.5 - In Exercises 5 and 6, follow the method of...Ch. 1.5 - In Exercises 7-12, describe all solutions of Ax =...Ch. 1.5 - In Exercises 7-12, describe all solutions of Ax =...Ch. 1.5 - In Exercises 7-12, describe all solutions of Ax =...Ch. 1.5 - In Exercises 7-12, describe all solutions of Ax =...Ch. 1.5 - In Exercises 7-12, describe all solutions of Ax =...Ch. 1.5 - In Exercises 7-12, describe all solutions of Ax =...Ch. 1.5 - Suppose the solution set of a certain system of...Ch. 1.5 - Suppose the solution set of a certain system of...Ch. 1.5 - Follow the method of Example 3 to describe the...Ch. 1.5 - As in Exercise 15, describe the solutions of the...Ch. 1.5 - Describe and compare the solution sets of x1 + 9x2...Ch. 1.5 - Describe and compare the solution sets of x1 3x2...Ch. 1.5 - In Exercises 19 and 20, find the parametric...Ch. 1.5 - In Exercises 19 and 20, find the parametric...Ch. 1.5 - In Exercises 21 and 22, find a parametric equation...Ch. 1.5 - In Exercises 21 and 22, find a parametric equation...Ch. 1.5 - a. A homogeneous equation is always consistent. b....Ch. 1.5 - a. If x is a nontrivial solution of Ax = 0, then...Ch. 1.5 - Prove the second part of Theorem 6: Let w be any...Ch. 1.5 - Suppose Ax = b has a solution. Explain why the...Ch. 1.5 - Suppose A is the 3 3 zero matrix (with all zero...Ch. 1.5 - If b 0, can the solution set of Ax = b be a plane...Ch. 1.5 - In Exercises 29-32, (a) does the equation Ax = 0...Ch. 1.5 - In Exercises 29-32, (a) does the equation Ax = 0...Ch. 1.5 - In Exercises 29-32, (a) does the equation Ax = 0...Ch. 1.5 - In Exercises 29-32, (a) does the equation Ax = 0...Ch. 1.5 - Given A = [2672139], find one nontrivial solution...Ch. 1.5 - Given A = [4681269], find one nontrivial solution...Ch. 1.5 - Construct a 3 3 nonzero matrix A such that the...Ch. 1.5 - Construct a 3 3 nonzero matrix A such that the...Ch. 1.5 - Construct a 2 2 matrix A such that the solution...Ch. 1.5 - Suppose A is a 3 3 matrix and y is a vector in 3...Ch. 1.5 - Let A be an m n matrix and let u be a vector in n...Ch. 1.5 - Let A be an m n matrix, and let u and v be...Ch. 1.6 - Suppose an economy has three sectors: Agriculture,...Ch. 1.6 - Consider the network flow studied in Example 2....Ch. 1.6 - Suppose an economy has only two sectors, Goods and...Ch. 1.6 - Find another set of equilibrium prices for the...Ch. 1.6 - Boron sulfide reacts violently with water to form...Ch. 1.6 - When solutions of sodium phosphate and barium...Ch. 1.6 - Alka-Seltzer contains sodium bicarbonate (NaHCO3)...Ch. 1.6 - The following reaction between potassium...Ch. 1.6 - Find the general flow pattern of the network shown...Ch. 1.6 - a. Find the general traffic pattern in the freeway...Ch. 1.6 - a. Find the general flow pattern in the network...Ch. 1.6 - Intersections in England are often constructed as...Ch. 1.7 - Let u = [324] , v = [617] , w = [052] , and z =...Ch. 1.7 - Suppose that {v1, v2, v3} is a linearly dependent...Ch. 1.7 - In Exercises 1-4, determine if the vectors are...Ch. 1.7 - In Exercises 1-4, determine if the vectors are...Ch. 1.7 - In Exercises 1-4, determine if the vectors are...Ch. 1.7 - In Exercises 1-4, determine if the vectors are...Ch. 1.7 - In Exercises 5-8, determine if the columns of the...Ch. 1.7 - In Exercises 5-8, determine if the columns of the...Ch. 1.7 - In Exercises 5-8, determine if the columns of the...Ch. 1.7 - In Exercises 5-8, determine if the columns of the...Ch. 1.7 - In Exercises 9 and 10, (a) for what values of h is...Ch. 1.7 - In Exercises 9 and 10, (a) for what values of h is...Ch. 1.7 - In Exercises 11-14, find the value(s) of h for...Ch. 1.7 - In Exercises 11-14, find the value(s) of h for...Ch. 1.7 - In Exercises 11-14, find the value(s) of h for...Ch. 1.7 - In Exercises 11-14, find the value(s) of h for...Ch. 1.7 - Determine by inspection whether the vectors in...Ch. 1.7 - Determine by inspection whether the vectors in...Ch. 1.7 - Determine by inspection whether the vectors in...Ch. 1.7 - Determine by inspection whether the vectors in...Ch. 1.7 - Determine by inspection whether the vectors in...Ch. 1.7 - Determine by inspection whether the vectors in...Ch. 1.7 - In Exercises 21 and 22, mark each statement True...Ch. 1.7 - In Exercises 21 and 22, mark each statement True...Ch. 1.7 - In Exercises 23-26, describe the possible echelon...Ch. 1.7 - In Exercises 23-26, describe the possible echelon...Ch. 1.7 - In Exercises 23-26, describe the possible echelon...Ch. 1.7 - In Exercises 23-26, describe the possible echelon...Ch. 1.7 - How many pivot columns must a 7 5 matrix have if...Ch. 1.7 - How many pivot columns must a 5 7 matrix have if...Ch. 1.7 - Construct 3 2 matrices A and B such that Ax = 0...Ch. 1.7 - a. Fill in the blank in the following statement:...Ch. 1.7 - Exercises 31 and 32 should be solved without...Ch. 1.7 - Exercises 31 and 32 should be solved without...Ch. 1.7 - Each statement in Exercises 33-38 is either true...Ch. 1.7 - Each statement in Exercises 33-38 is either true...Ch. 1.7 - Each statement in Exercises 33-38 is either true...Ch. 1.7 - Each statement in Exercises 33-38 is either true...Ch. 1.7 - Each statement in Exercises 33-38 is either true...Ch. 1.7 - Each statement in Exercises 33-38 is either true...Ch. 1.7 - Suppose A is an m n matrix with the property that...Ch. 1.7 - Suppose an m n matrix A has n pivot columns....Ch. 1.7 - [M] In Exercises 41 and 42, use as many columns of...Ch. 1.7 - [M] In Exercises 41 and 42, use as many columns of...Ch. 1.8 - Suppose T : 5 2 and T(x) = Ax for some matrix A...Ch. 1.8 - A=[1001] Give a geometric description of the...Ch. 1.8 - The line segment from 0 to a vector u is the set...Ch. 1.8 - Let A=[2002], and define T : 22 by T(x) = Ax. Find...Ch. 1.8 - Let A=[.5000.5000.5], u=[104], and v=[abc]. Define...Ch. 1.8 - In Exercises 3-6, with T defined by T(x) = Ax,...Ch. 1.8 - In Exercises 3-6, with T defined by T(x) = Ax,...Ch. 1.8 - In Exercises 3-6, with T defined by T(x) = Ax,...Ch. 1.8 - In Exercises 3-6, with T defined by T(x) = Ax,...Ch. 1.8 - Let A be a 6 5 matrix. What must a and b be in...Ch. 1.8 - How many rows and columns must a matrix A have in...Ch. 1.8 - For Exercises 9 and 10, find all x in 4 that are...Ch. 1.8 - For Exercises 9 and 10, find all x in 4 that are...Ch. 1.8 - Let b=[110], and let A be the matrix in Exercise...Ch. 1.8 - Let b=[1314]. and let A be the matrix in Exercise...Ch. 1.8 - In Exercises 13-16, use a rectangular coordinate...Ch. 1.8 - In Exercises 13-16, use a rectangular coordinate...Ch. 1.8 - In Exercises 13-16, use a rectangular coordinate...Ch. 1.8 - In Exercises 13-16, use a rectangular coordinate...Ch. 1.8 - Let T : 2 2 be a linear transformation that maps...Ch. 1.8 - The figure shows vectors u, v, and w, along with...Ch. 1.8 - Let e1=[10], e2=[01], y1=[25], and y2=[16], and...Ch. 1.8 - Let x=[x1x2], v1=[25], and v2=[73], and let T : 2 ...Ch. 1.8 - In Exercises 21 and 22, mark each statement True...Ch. 1.8 - In Exercises 21 and 22, mark each statement True...Ch. 1.8 - Let T : 2 2 be the linear transformation that...Ch. 1.8 - Suppose vectors v1, . . . , vp span n, and let T :...Ch. 1.8 - Prob. 25ECh. 1.8 - Let u and v be linearly independent vectors in 3,...Ch. 1.8 - Prob. 27ECh. 1.8 - Let u and v be vectors in n. It can be shown that...Ch. 1.8 - Define f : by f(x) = mx + b. a. Show that f is...Ch. 1.8 - An affine transformation T : n m has the form...Ch. 1.8 - Let T : n m be a linear transformation, and let...Ch. 1.8 - In Exercises 32-36, column vectors are written as...Ch. 1.8 - In Exercises 32-36, column vectors are written as...Ch. 1.8 - In Exercises 32-36, column vectors are written as...Ch. 1.8 - In Exercises 32-36, column vectors are written as...Ch. 1.8 - In Exercises 32-36, column vectors are written as...Ch. 1.8 - [M] In Exercises 37 and 38, the given matrix...Ch. 1.8 - [M] In Exercises 37 and 38, the given matrix...Ch. 1.8 - [M] Let b=[7597] and let A be the matrix in...Ch. 1.8 - [M] Let b=[77135] and let A be the matrix in...Ch. 1.9 - Let T : 2 2 be the transformation that first...Ch. 1.9 - Suppose A is a 7 5 matrix with 5 pivots. Let T(x)...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - In Exercises 1-10, assume that T is a linear...Ch. 1.9 - A linear transformation T : 2 2 first reflects...Ch. 1.9 - Show that the transformation in Exercise 8 is...Ch. 1.9 - Let T : 2 be the linear transformation such that...Ch. 1.9 - Let T : 2 2 be a linear transformation with...Ch. 1.9 - In Exercises 15 and 16, fill in the missing...Ch. 1.9 - In Exercises 15 and 16, fill in the missing...Ch. 1.9 - In Exercises 17-20, show that T is a linear...Ch. 1.9 - In Exercises 17-20, show that T is a linear...Ch. 1.9 - In Exercises 17-20, show that T is a linear...Ch. 1.9 - In Exercises 17-20, show that T is a linear...Ch. 1.9 - Let T : 2 2 be a linear transformation such that...Ch. 1.9 - Let T : 2 3 be a linear transformation such that...Ch. 1.9 - In Exercises 23 and 24, mark each statement True...Ch. 1.9 - a. Not every linear transformation from n to m is...Ch. 1.9 - In Exercises 25-28, determine if the specified...Ch. 1.9 - In Exercises 25-28, determine if the specified...Ch. 1.9 - In Exercises 25-28, determine if the specified...Ch. 1.9 - In Exercises 25-28, determine if the specified...Ch. 1.9 - In Exercises 29 and 30, describe the possible...Ch. 1.9 - In Exercises 29 and 30, describe the possible...Ch. 1.9 - Let T : n m be a linear transformation, with A...Ch. 1.9 - Let T : n m be a linear transformation, with A...Ch. 1.9 - Verify the uniqueness of A in Theorem 10. Let T :...Ch. 1.9 - Why is the question Is the linear transformation T...Ch. 1.9 - If a linear transformation T : n m maps n onto m,...Ch. 1.9 - Let S : p n and T : n m be linear...Ch. 1.9 - [M] In Exercises 37-40, let T be the linear...Ch. 1.9 - [M] In Exercises 37-40, let T be the linear...Ch. 1.9 - [M] In Exercises 37-40, let T be the linear...Ch. 1.9 - [M] In Exercises 37-40, let T be the linear...Ch. 1.10 - Find a matrix A and vectors x and b such that the...Ch. 1.10 - The container of a breakfast cereal usually lists...Ch. 1.10 - One serving of Post Shredded Wheat supplies 160...Ch. 1.10 - After taking a nutrition class, a big Annies Mac...Ch. 1.10 - The Cambridge Diet supplies .8 g of calcium per...Ch. 1.10 - In Exercises 5-8, write a matrix equation that...Ch. 1.10 - In Exercises 5-8, write a matrix equation that...Ch. 1.10 - In Exercises 5-8, write a matrix equation that...Ch. 1.10 - In Exercises 5-8, write a matrix equation that...Ch. 1.10 - In a certain region, about 7% of a citys...Ch. 1.10 - In a certain region, about 6% of a citys...Ch. 1.10 - In 2012 the population of California was...Ch. 1.10 - [M] Budget Rent A Car in Wichita. Kansas, has a...Ch. 1.10 - [M] Let M and xo be as in Example 3. a. Compute...Ch. 1.10 - [M] Study how changes in boundary temperatures on...Ch. 1 - Mark each statement True or False. Justify each...Ch. 1 - Let a and b represent real numbers. Describe the...Ch. 1 - The solutions (x, y, Z) of a single linear...Ch. 1 - Suppose the coefficient matrix of a linear system...Ch. 1 - Determine h and k such that the solution set of...Ch. 1 - Consider the problem of determining whether the...Ch. 1 - Consider the problem of determining whether the...Ch. 1 - Describe the possible echelon forms of the matrix...Ch. 1 - Prob. 9SECh. 1 - Let a1, a2 and b be the vectors in 2 shown in the...Ch. 1 - Construct a 2 3 matrix A, not in echelon form,...Ch. 1 - Construct a 2 3 matrix A, not in echelon form,...Ch. 1 - Write the reduced echelon form of a 3 3 matrix A...Ch. 1 - Determine the value(s) of a such that...Ch. 1 - In (a) and (b), suppose the vectors are linearly...Ch. 1 - Use Theorem 7 in Section 1.7 to explain why the...Ch. 1 - Explain why a set {v1, v2, v3, v4} in 5 must be...Ch. 1 - Suppose {v1, v2} is a linearly independent set in...Ch. 1 - Suppose v1, v2, v3 are distinct points on one line...Ch. 1 - Let T : n m be a linear transformation, and...Ch. 1 - Let T : 3 3 be the linear transformation that...Ch. 1 - Let A be a 3 3 matrix with the property that the...Ch. 1 - A Givens rotation is a linear transformation from...Ch. 1 - The following equation describes a Givens rotation...Ch. 1 - A large apartment building is to be built using...
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- R denotes the field of real numbers, Q denotes the field of rationals, and Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures. Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent (a) Let vi = x, V2 = list in R[x] 3. (b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4) is a basis of R[x] 3. [8] [6] (c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a linear map. [6] (d) Write down the matrix for the map ƒ defined in (c) with respect to the basis (2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3. [5]arrow_forwardQuestion 4 (a) The following matrices represent linear maps on R² with respect to an orthonormal basis: = [1/√5 2/√5 [2/√5 -1/√5] " [1/√5 2/√5] A = B = [2/√5 1/√5] 1 C = D = = = [ 1/3/5 2/35] 1/√5 2/√5 -2/√5 1/√5' For each of the matrices A, B, C, D, state whether it represents a self-adjoint linear map, an orthogonal linear map, both, or neither. (b) For the quadratic form q(x, y, z) = y² + 2xy +2yz over R, write down a linear change of variables to u, v, w such that q in these terms is in canonical form for Sylvester's Law of Inertia. [6] [4]arrow_forwardpart b pleasearrow_forward
- Question 5 (a) Let a, b, c, d, e, ƒ Є K where K is a field. Suppose that the determinant of the matrix a cl |df equals 3 and the determinant of determinant of the matrix a+3b cl d+3e f ГЪ e [ c ] equals 2. Compute the [5] (b) Calculate the adjugate Adj (A) of the 2 × 2 matrix [1 2 A = over R. (c) Working over the field F3 with 3 elements, use row and column operations to put the matrix [6] 0123] A = 3210 into canonical form for equivalence and write down the canonical form. What is the rank of A as a matrix over F3? 4arrow_forwardQuestion 2 In this question, V = Q4 and - U = {(x, y, z, w) EV | x+y2w+ z = 0}, W = {(x, y, z, w) € V | x − 2y + w − z = 0}, Z = {(x, y, z, w) € V | xyzw = 0}. (a) Determine which of U, W, Z are subspaces of V. Justify your answers. (b) Show that UW is a subspace of V and determine its dimension. (c) Is VU+W? Is V = UW? Justify your answers. [10] [7] '00'arrow_forwardTools Sign in Different masses and Indicated velocities Rotational inert > C C Chegg 39. The balls shown have different masses and speeds. Rank the following from greatest to least: 2.0 m/s 8.5 m/s 9.0 m/s 12.0 m/s 1.0 kg A 1.2 kg B 0.8 kg C 5.0 kg D C a. The momenta b. The impulses needed to stop the balls Solved 39. The balls shown have different masses and speeds. | Chegg.com Images may be subject to copyright. Learn More Share H Save Visit > quizlet.com%2FBoyE3qwOAUqXvw95Fgh5Rw.jpg&imgrefurl=https%3A%2F%2Fquizlet.com%2F529359992%2Fc. Xarrow_forward
- 2. What is the total length of the shortest path that goes from (0,4) to a point on the x-axis, then to a point on the line y = 6, then to (18.4)?arrow_forwardموضوع الدرس Prove that Determine the following groups Homz(QZ) Hom = (Q13,Z) Homz(Q), Hom/z/nZ, Qt for neN- (2) Every factor group of adivisible group is divisble. • If R is a Skew ficald (aring with identity and each non Zero element is invertible then every R-module is free.arrow_forwardI have ai answers but incorrectarrow_forward
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