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For each of the following statements, choose whether it is True or False. Correct answers are worth one point each. (a) If A is a k x k matrix and A is not invertible, then λ = 0 is an eigenvalue of A. True (b) If B is a 4 x 4 real matrix, then B has at least one real eigenvalue. False (c) The set v-{[₁₂ V 21 +222 +23] is a two-dimensional subspace of R³. (d) If {v₁, 2,. Un} is an orthonormal basis of R" and w E R", then, for 1≤j≤n, the coordinate of u with respect to ; is given by the dot True product w.vj. ...9 True x1 + x2 x2 + x3 : 1, 2, 3 ER True (e) If A is a k x k diagonalizable matrix then A is invertible. False False (f) If A is a 5 x 5 real matrix, then A has at least one real eigenvalue. If A is a 5 x 6 real matrix, then ATA is diagonalizable. (h) The set (mm) Plml;n(1) + n(?) - 0} is a (vector) subspace of the

For each of the following statements, choose whether it is True or False. Correct answers are worth one point each. (a) If A is a k x k matrix and A is not invertible, then λ = 0 is an eigenvalue of A. True (b) If B is a 4 x 4 real matrix, then B has at least one real eigenvalue. False (c) The set v-{[₁₂ V 21 +222 +23] is a two-dimensional subspace of R³. (d) If {v₁, 2,. Un} is an orthonormal basis of R" and w E R", then, for 1≤j≤n, the coordinate of u with respect to ; is given by the dot True product w.vj. ...9 True x1 + x2 x2 + x3 : 1, 2, 3 ER True (e) If A is a k x k diagonalizable matrix then A is invertible. False False (f) If A is a 5 x 5 real matrix, then A has at least one real eigenvalue. If A is a 5 x 6 real matrix, then ATA is diagonalizable. (h) The set (mm) Plml;n(1) + n(?) - 0} is a (vector) subspace of the

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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For each of the following statements, choose whether it is True or False. Correct answers are worth one point each. (a) If A is a k x k matrix and A is not invertible, then λ = 0 is an eigenvalue of A. True (b) If B is a 4 x 4 real matrix, then B has at least one real eigenvalue. False (c) The set -{[ V = True is a two-dimensional subspace of R³. False V (d) If {v₁, V2,...,n} is an orthonormal basis of R" and w E R", then, for 1≤j≤n, the coordinate of u with respect to ū, is given by the dot product w.vj. True True x1 + x2 x2 + x3 x1 + 2x2 + x3 (e) If A is a k x k diagonalizable matrix then A is invertible. False (f) If A is a 5 x 5 real matrix, then A has at least one real eigenvalue. . V : x1, x2, x3 ER v (g) If A is a 5 x 6 real matrix, then ATA is diagonalizable. v 3 (yector) subspace of the
Transcribed Image Text:For each of the following statements, choose whether it is True or False. Correct answers are worth one point each. (a) If A is a k x k matrix and A is not invertible, then λ = 0 is an eigenvalue of A. True (b) If B is a 4 x 4 real matrix, then B has at least one real eigenvalue. False (c) The set -{[ V = True is a two-dimensional subspace of R³. False V (d) If {v₁, V2,...,n} is an orthonormal basis of R" and w E R", then, for 1≤j≤n, the coordinate of u with respect to ū, is given by the dot product w.vj. True True x1 + x2 x2 + x3 x1 + 2x2 + x3 (e) If A is a k x k diagonalizable matrix then A is invertible. False (f) If A is a 5 x 5 real matrix, then A has at least one real eigenvalue. . V : x1, x2, x3 ER v (g) If A is a 5 x 6 real matrix, then ATA is diagonalizable. v 3 (yector) subspace of the
(h) The set {p(x) = R[x]: p(1) + p(2) = 0} is a (vector) subspace of the (vector) space of real polynomials, R[x]. True (i) If A is a 7 x 11 real matrix, then A has at most 7 singular values. False (j) The kernel of the linear map T B-L x + y X-Z 2x + y - z 3 is a two-dimensional subspace of R³. False
Transcribed Image Text:(h) The set {p(x) = R[x]: p(1) + p(2) = 0} is a (vector) subspace of the (vector) space of real polynomials, R[x]. True (i) If A is a 7 x 11 real matrix, then A has at most 7 singular values. False (j) The kernel of the linear map T B-L x + y X-Z 2x + y - z 3 is a two-dimensional subspace of R³. False
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