Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
5th Edition
ISBN: 9780134689531
Author: Lee Johnson, Dean Riess, Jimmy Arnold
Publisher: PEARSON
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Chapter 3 Solutions
Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
Ch. 3.1 - Prob. 1ECh. 3.1 - Prob. 2ECh. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Prob. 4ECh. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Prob. 6ECh. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Prob. 10E
Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - In Exercises 12-17, interpret the subset W of R2...Ch. 3.1 - In Exercises 12-17, interpret the subset W of R2...Ch. 3.1 - In Exercises 12-17, interpret the subset W of R2...Ch. 3.1 - In Exercises 12-17, interpret the subset W of R2...Ch. 3.1 - In Exercises 12-17, interpret the subset W of R2...Ch. 3.1 - Prob. 17ECh. 3.1 - Prob. 18ECh. 3.1 - In Exercises 18-21, Interpret the subset W of R3...Ch. 3.1 - In Exercises 18-21, Interpret the subset W of R3...Ch. 3.1 - Prob. 21ECh. 3.1 - In Exercises 22-26, give a set-theoretic...Ch. 3.1 - In Exercises 22-26, give a set theoretic...Ch. 3.1 - In Exercises 22-26, give a set theoretic...Ch. 3.1 - In Exercises 22-26, give a settheoretic...Ch. 3.1 - In Exercises 22-26, give a settheoretic...Ch. 3.1 - In Exercises 27-30, give a settheoretic...Ch. 3.1 - In Exercises 27-30, give a set theoretic...Ch. 3.1 - In Exercises 27-30, give a set theoretic...Ch. 3.1 - In Exercises 27-30, give a settheoretic...Ch. 3.2 - In Exercise 1-8, W is a subset of R2 consisting of...Ch. 3.2 - In Exercise 1-8, W is a subset of R2 consisting of...Ch. 3.2 - In Exercise 1-8, W is a subset of R2 consisting of...Ch. 3.2 - In Exercise 1-8, W is a subset of R2 consisting of...Ch. 3.2 - In Exercise 1-8, W is a subset of R2 consisting of...Ch. 3.2 - In Exercise 1-8, W is a subset of R2 consisting of...Ch. 3.2 - In Exercise 1-8, W is a subset of R2 consisting of...Ch. 3.2 - In Exercise 1-8, W is a subset of R2 consisting of...Ch. 3.2 - In Exercise 9-17, W is a subset of R3 consisting...Ch. 3.2 - In Exercise 9-17, W is a subset of R3 consisting...Ch. 3.2 - In Exercise 9-17, W is a subset of R3 consisting...Ch. 3.2 - In Exercise 9-17, W is a subset of R3 consisting...Ch. 3.2 - In Exercise 9-17, W is a subset of R3 consisting...Ch. 3.2 - In Exercise 9-17, W is a subset of R3 consisting...Ch. 3.2 - In Exercise 9-17, W is a subset of R3 consisting...Ch. 3.2 - In Exercise 9-17, W is a subset of R3 consisting...Ch. 3.2 - In Exercise 9-17, W is a subset of R3 consisting...Ch. 3.2 - Let abe a fixed vector in R3, and define Wto be...Ch. 3.2 - Let W be the subspace defined in Exercise 18,...Ch. 3.2 - Let W be the subspace defined in Exercise 18,...Ch. 3.2 - Let a and b be fixed vectors in R3, and let W be...Ch. 3.2 - In Exercises 22-25, W is the subspace of R3...Ch. 3.2 - Prob. 26ECh. 3.2 - In R2, suppose that scalar multiplication were...Ch. 3.2 - Let W=x:x=[x1x2],x20. In the statement of Theorem...Ch. 3.2 - In R3, a line through the origin is the set of all...Ch. 3.2 - If U and V are subsets of Rn, then the set U+V is...Ch. 3.2 - Let U and V be subspaces of Rn. Prove that the...Ch. 3.2 - Let U and V be the subspaces of R3 defined by...Ch. 3.2 - Let U and V be the subspaces of Rn a) Show that...Ch. 3.2 - Prob. 34ECh. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 111 refer to the vectors in Eq. (14)....Ch. 3.3 - Exercises 12-19 refer to the vectors in Eq. 15....Ch. 3.3 - Exercises 12-19 refer to the vectors in Eq. 15....Ch. 3.3 - Exercises 12-19 refer to the vectors in Eq. 15....Ch. 3.3 - Exercise 1219 refer to the vector in Eq.15....Ch. 3.3 - Exercise 1219 refer to the vector in Eq.15....Ch. 3.3 - Exercise 1219 refer to the vector in Eq.15....Ch. 3.3 - Exercise 1219 refer to the vector in Eq.15....Ch. 3.3 - Exercise 1219 refer to the vector in Eq.15....Ch. 3.3 - Let S be the set given in Exercise 14. For each...Ch. 3.3 - Repeat Exercise 20. for the set S given in...Ch. 3.3 - Determine which of the vectors listed in Eq. (14)...Ch. 3.3 - Determine which of the vectors listed in Eq. (14)...Ch. 3.3 - Determine which of the vectors listed in Eq. (15)...Ch. 3.3 - Determine which of the vectors listed in Eq. (15)...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - In Exercises 26-27, give an algebraic...Ch. 3.3 - In Exercises 26-27, give an algebraic...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - In Exercise 2637, give an algebraic specification...Ch. 3.3 - Let A be the matrix given in Exercise 26. aFor...Ch. 3.3 - Repeat Exercise 38 for the matrix given in...Ch. 3.3 - Let A be the matrix given in Exercise 34. aFor...Ch. 3.3 - Repeat Exercise 40 for the given matrix in...Ch. 3.3 - Let...Ch. 3.3 - let W={x=[x1x2x3]:3x14x2+2x3=0}. Exhibit a (13)...Ch. 3.3 - Let S be the set of vectors given in Exercise 16....Ch. 3.3 - Let S be the set of vectors given in Exercise 17....Ch. 3.3 - In Exercises 46-49, use the technique illustrated...Ch. 3.3 - In Exercises 46-49, use the technique illustrated...Ch. 3.3 - In Exercises 46-49, use the technique illustrated...Ch. 3.3 - In Exercises 46-49, use the technique illustrated...Ch. 3.3 - Identify the range and the null space for each of...Ch. 3.3 - Prob. 51ECh. 3.3 - Let A be an (mr) matrix and B an (rn) matrix....Ch. 3.3 - Prob. 53ECh. 3.3 - Prob. 54ECh. 3.4 - In Exercises 18, let W be the subspace of R4...Ch. 3.4 - In Exercises 18, let W be the subspace of R4...Ch. 3.4 - In Exercises 18, let W be the subspace of R4...Ch. 3.4 - In Exercises 18, let W be the subspace of R4...Ch. 3.4 - In Exercises 18, let W be the subspace of R4...Ch. 3.4 - In Exercises 18, let W be the subspace of R4...Ch. 3.4 - In Exercises 18, let W be the subspace of R4...Ch. 3.4 - In Exercises 18, let W be the subspace of R4...Ch. 3.4 - Let W be the subspace described in Exercise 1. For...Ch. 3.4 - Let W be the subspace described in Exercise 2. For...Ch. 3.4 - In Exercises 11-16: a Find a matrix B in reduced...Ch. 3.4 - In Exercises 11-16: a Find a matrix B in reduced...Ch. 3.4 - In Exercises 11-16: a Find a matrix B in reduced...Ch. 3.4 - In Exercises 11-16: a Find a matrix B in reduced...Ch. 3.4 - In Exercises 1116: a) Find a matrix B in reduced...Ch. 3.4 - In Exercises 1116: a) Find a matrix B in reduced...Ch. 3.4 - Repeat Exercise 17 for the matrix given in...Ch. 3.4 - Repeat Exercise 17 for the matrix given in...Ch. 3.4 - Repeat Exercise 17 for the matrix given in...Ch. 3.4 - In Exercise 21-24 for the given set S: a Find a...Ch. 3.4 - In Exercise 21-24 for the given set S: a Find a...Ch. 3.4 - In Exercise 21-24 for the given set S: a Find a...Ch. 3.4 - In Exercise 21-24 for the given set S: a Find a...Ch. 3.4 - Find a basis for the null space of each of the...Ch. 3.4 - Find a basis for the range of each matrix in...Ch. 3.4 - Let S={v1,v2,v3} where v1=[121], v2=[111], and...Ch. 3.4 - Let S={v1,v2,v3}, where v1=[10], v2=[01] and...Ch. 3.4 - Let S={v1,v2,v3,v4}, where v1=[121],...Ch. 3.4 - Let B={v1,v2,v3} be a set of linearly independent...Ch. 3.4 - Let B={v1,v2,v3} be a subset of R3 such that...Ch. 3.4 - In Exercises 32-35, determine whether the given...Ch. 3.4 - In Exercises 32-35, determine whether the given...Ch. 3.4 - In Exercises 32-35, determine whether the given...Ch. 3.4 - In Exercises 32-35, determine whether the given...Ch. 3.4 - Find vector w in R3 such that w is not a linear...Ch. 3.4 - Prob. 37ECh. 3.4 - Prob. 38ECh. 3.4 - Recalling Exercises 38, prove that every basis for...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - Exercises 1-14 refer to the vectors in 15 u1=[11],...Ch. 3.5 - In Exercises 15-20, W is a subspace of R4...Ch. 3.5 - In Exercises 15-20, W is a subspace of R4...Ch. 3.5 - In Exercises 15-20, W is a subspace of R4...Ch. 3.5 - In Exercises 15-20, W is a subspace of R4...Ch. 3.5 - In Exercises 15-20, W is a subspace of R4...Ch. 3.5 - In Exercises 15-20, W is a subspace of R4...Ch. 3.5 - In Exercises 21-24, find a basis for N(A) and give...Ch. 3.5 - In Exercise 21-24, find a basis for N(A) and give...Ch. 3.5 - In Exercise 21-24, find a basis for N(A) and give...Ch. 3.5 - In Exercise 21-24, find a basis for N(A) and give...Ch. 3.5 - In Exercise 25-26, find a basis for R(A) and give...Ch. 3.5 - In Exercise 25-26, find a basis for R(A) and give...Ch. 3.5 - Let W be a subspace, and let S be a spanning set...Ch. 3.5 - Let W the subset of R4 defined by W={x:vTx=0}...Ch. 3.5 - Let W be the subspace of R4 defined by...Ch. 3.5 - Let W be a nonzero subspace of Rn. Show that W has...Ch. 3.5 - Suppose that {u1,u2,,up} is a basis for a subspace...Ch. 3.5 - Let U and V be subspace of Rn, and suppose that U...Ch. 3.5 - For each of the following, determine the largest...Ch. 3.5 - If A is a (34) matrix, prove that the columns of A...Ch. 3.5 - If A is a (43) matrix, prove that the rows of A...Ch. 3.5 - Let A be an (mn) matrix. Prove that rank (A)m and...Ch. 3.5 - Let A be an (23) matrix with rank 2. Show that the...Ch. 3.5 - Let A be an (34) matrix with nullity 1. Prove that...Ch. 3.5 - Prove that an (nn) matrix is nonsingular if and...Ch. 3.5 - Prob. 40ECh. 3.5 - Prob. 41ECh. 3.5 - Prob. 42ECh. 3.6 - In Exercises 14, verify that u1,u2,u3 is an...Ch. 3.6 - In Exercises 14, verify that u1,u2,u3 is an...Ch. 3.6 - In Exercises 14, verify that u1,u2,u3 is an...Ch. 3.6 - In Exercises 14, verify that u1,u2,u3 is an...Ch. 3.6 - In Exercises 58, find values a, b, and c such that...Ch. 3.6 - In Exercises 58, find values a, b, and c such that...Ch. 3.6 - In Exercises 58, find values a, b, and c such that...Ch. 3.6 - In Exercises 58, find values a, b, and c such that...Ch. 3.6 - In Exercises 912, express the given vector v in...Ch. 3.6 - In Exercises 912, express the given vector v in...Ch. 3.6 - In Exercises 912, express the given vector v in...Ch. 3.6 - In Exercises 912, express the given vector v in...Ch. 3.6 - In Exercises 1318, use the Gram-Schmidt process to...Ch. 3.6 - In Exercises 1318, use the Gram-Schmidt process to...Ch. 3.6 - In Exercises 1318, use the Gram-Schmidt process to...Ch. 3.6 - In Exercises 1318, use the Gram-Schmidt process to...Ch. 3.6 - In Exercises 1318, use the Gram-Schmidt process to...Ch. 3.6 - In Exercises 1318, use the Gram-Schmidt process to...Ch. 3.6 - In Exercises 19 and 20, find a basis for the null...Ch. 3.6 - In Exercises 19 and 20, find a basis for the null...Ch. 3.6 - Argue that any set of four or more nonzero vectors...Ch. 3.6 - Let S=u1,u2,u3 be an orthogonal set of nonzero...Ch. 3.6 - Prob. 23ECh. 3.6 - Prob. 24ECh. 3.6 - The triangle inequality. Let x and y be vectors in...Ch. 3.6 - Let x and y be vectors in Rn. Prove that...Ch. 3.6 - Prob. 27ECh. 3.6 - Let B=u1,u2,.........,up be an orthonormal basis...Ch. 3.7 - Define T:R2R2 by T([x1x2])=[2x13x2x1+x2] Find each...Ch. 3.7 - Define T:R2R2 by T(x)=Ax, where A=[1133] Find each...Ch. 3.7 - Let T:R2R2 be the linear transformation defined by...Ch. 3.7 - Let T:R2R2 be the function defined in Exercise 1....Ch. 3.7 - Let T:R2R2 be the function given in Exercise 1....Ch. 3.7 - Let T be the linear transformation given in...Ch. 3.7 - Let T be the linear transformation given in...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - In Exercise 817, determine whether the function F...Ch. 3.7 - Let W be the subspace of R3 defined by...Ch. 3.7 - Let T:R2R3 be a linear transformation such that...Ch. 3.7 - Let T:R2R2 be a linear transformation such that...Ch. 3.7 - In Exercise 21-24, the action of a linear...Ch. 3.7 - In Exercise 21-24, the action of a linear...Ch. 3.7 - In Exercise 21-24, the action of a linear...Ch. 3.7 - In Exercise 21-24, the action of a linear...Ch. 3.7 - In Exercise 25-30, a linear transformation T is...Ch. 3.7 - In Exercise 25-30, a linear transformation T is...Ch. 3.7 - In Exercise 25-30, a linear transformation T is...Ch. 3.7 - In Exercise 25-30, a linear transformation T is...Ch. 3.7 - In Exercise 25-30, a linear transformation T is...Ch. 3.7 - In Exercise 25-30, a linear transformation T is...Ch. 3.7 - Let a be a real number, and define f:RR by f(x)=ax...Ch. 3.7 - Let T:RR be a linear transformation, and suppose...Ch. 3.7 - Let T:R2R2 be the function that maps each point in...Ch. 3.7 - Let T:R2R2 be the function that maps each point in...Ch. 3.7 - Let V and W be subspaces, and let F:VW and G:VW be...Ch. 3.7 - Let F:R3R2 and G:R3R2 defined by...Ch. 3.7 - Let V and W be subspaces, and let T:VW be linear...Ch. 3.7 - Let T:R3R2 be the linear transformation defined in...Ch. 3.7 - Let U,V and W be subspaces, and let F:UV and G:VW...Ch. 3.7 - Let F:R3R2 and G:R2R3 be linear transformations...Ch. 3.7 - Let B be an (mn) matrix, and let T:RnRm be defined...Ch. 3.7 - Let F:RnRp and G:RpRm be linear transformations,...Ch. 3.7 - I:RnRm be the identity transformation. Determine...Ch. 3.7 - Prob. 44ECh. 3.7 - Prob. 45ECh. 3.7 - Prob. 46ECh. 3.7 - Prob. 47ECh. 3.7 - Prob. 48ECh. 3.7 - Exercises 4549 are based on the optional material....Ch. 3.8 - In Exercise 1-6, find all vectors x that minimize...Ch. 3.8 - In Exercise 1-6, find all vectors x that minimize...Ch. 3.8 - In Exercise 1-6, find all vectors x that minimize...Ch. 3.8 - In Exercise 1-6, find all vectors x that minimize...Ch. 3.8 - In Exercise 1-6, find all vectors x that minimize...Ch. 3.8 - In Exercise 1-6, find all vectors x that minimize...Ch. 3.8 - In Exercises 7-10, find the least-squares linear...Ch. 3.8 - Prob. 8ECh. 3.8 - Prob. 9ECh. 3.8 - Prob. 10ECh. 3.8 - Prob. 11ECh. 3.8 - In Exercises 11-14, find the least-squares...Ch. 3.8 - Prob. 13ECh. 3.8 - Prob. 14ECh. 3.8 - Consider the following table of data:...Ch. 3.8 - Prob. 16ECh. 3.8 - Prob. 17ECh. 3.8 - Prob. 18ECh. 3.9 - Prob. 1ECh. 3.9 - Prob. 2ECh. 3.9 - Prob. 3ECh. 3.9 - Prob. 4ECh. 3.9 - Exercise 116 refers to the following subspaces: b)...Ch. 3.9 - Prob. 6ECh. 3.9 - Exercise 116 refers to the following subspaces: c)...Ch. 3.9 - Exercise 116 refers to the following subspaces: b)...Ch. 3.9 - Prob. 9ECh. 3.9 - Prob. 10ECh. 3.9 - Prob. 11ECh. 3.9 - Prob. 12ECh. 3.9 - Prob. 13ECh. 3.9 - Prob. 14ECh. 3.9 - Prob. 15ECh. 3.9 - Prob. 16ECh. 3.9 - Prob. 17ECh. 3.SE - Let W={X:X=[x1x2],x1x2=0} Verify that W satisfies...Ch. 3.SE - 2. Let W={x:x=[x1x2],x10,x20}. Verify that W...Ch. 3.SE - Let A=[211141221] and W={x:x=[x1x2x3],Ax=3x}. a...Ch. 3.SE - If S={[112],[213]} And T={[105],[017],[321]}, Then...Ch. 3.SE - 5. Let A=[112322541107] a Reduce the matrix A to...Ch. 3.SE - 6. Let S={v1,v2,v3}, where v1=[111], v2=[121], and...Ch. 3.SE - Let A be an (mn) matrix defined by...Ch. 3.SE - In a)-c), use the given information to determine...Ch. 3.SE - Prob. 9SECh. 3.SE - Let B=x1,x2 be a basis for R2 and let T:R2R2 be a...Ch. 3.SE - Let b=[ab], and suppose that T:R3R2 is linear...Ch. 3.SE - In Exercise 12-18, b=[a,b,c,d]T, T:R6R4 is a...Ch. 3.SE - In Exercise 12-18, b=[a,b,c,d]T, T:R6R4 is a...Ch. 3.SE - In Exercise 12-18, b=[a,b,c,d]T, T:R6R4 is a...Ch. 3.SE - In Exercise 12-18, b=[a,b,c,d]T, T:R6R4 is a...Ch. 3.SE - In Exercises 12-18, b=[a,b,c,d]T, T:R6R4 is a...Ch. 3.SE - In Exercise 12-18, b=[a,b,c,d]T, T:R6R4 is a...Ch. 3.SE - In Exercise 12-18, b=[a,b,c,d]T, T:R6R4 is a...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In Exercises 1-12, answer true or false. Justify...Ch. 3.CE - In exercises 13-23, give a brief answer. Let W be...Ch. 3.CE - In exercises 13-23, give a brief answer. Explain...Ch. 3.CE - In exercises 13-23, give a brief answer. If B={x1,...Ch. 3.CE - In exercises 13-23, give a brief answer. Let W be...Ch. 3.CE - In exercises 13-23, give a brief answer. Let...Ch. 3.CE - In exercises 13-23, give a brief answer. Let u be...Ch. 3.CE - Let V and W be subspaces of Rn such that VW={} and...Ch. 3.CE - In exercises 13-23, give a brief answer. A linear...Ch. 3.CE - If T:RnRm is a linear transformation, then show...Ch. 3.CE - Let T:RnRn be a linear transformation, and suppose...Ch. 3.CE - Let T:RnRm be a linear transformation with nullity...
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