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(a) Label the following statements as true or false. Suppose that B = {x1, X2, ..., xn} and ß' = {x'1, x'2, ordered bases for a vector space and Q is the change of coordinate matrix that changes B'-coordinates into B-coordinates. Then the jth column of Q is •, = {x′1, x′2, ..., x'n} are [xj] B₁². Br (b) (c) Every change of coordinate matrix is invertible. Let T be a linear operator on a finite-dimensional vector space V, let B and ß′ be ordered bases for V, and let Q be the change of coordinate matrix that changes 3'-coordinates into ß-coordinates. Then [T] = Q[T] µQ¯¹. B

(a) Label the following statements as true or false. Suppose that B = {x1, X2, ..., xn} and ß' = {x'1, x'2, ordered bases for a vector space and Q is the change of coordinate matrix that changes B'-coordinates into B-coordinates. Then the jth column of Q is •, = {x′1, x′2, ..., x'n} are [xj] B₁². Br (b) (c) Every change of coordinate matrix is invertible. Let T be a linear operator on a finite-dimensional vector space V, let B and ß′ be ordered bases for V, and let Q be the change of coordinate matrix that changes 3'-coordinates into ß-coordinates. Then [T] = Q[T] µQ¯¹. B

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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1. Label the following statements as true or false. (a) (b) (c) (d) (e) Suppose that B = {x₁, x2, ..., xn} and ß′ = {x'1, x′2, ……., x'n} are ordered bases for a vector space and Q is the change of coordinate matrix that changes B'-coordinates into B-coordinates. Then the jth column of Q is [xj] ¹² Every change of coordinate matrix is invertible. Let T be a linear operator on a finite-dimensional vector space V, let B and B' be ordered bases for V, and let Q be the change of coordinate matrix that changes 3-coordinates into B-coordinates. Then [T] = Q[T] „Q¯¹. В The matrices A, B = Mnxn (F) are called similar if B = Q AQ for some QE Mnxn (F). Let T be a linear operator on a finite-dimensional vector space V. Then for any ordered bases 3 and y for V, [T] is similar to [T] B 'y
Transcribed Image Text:1. Label the following statements as true or false. (a) (b) (c) (d) (e) Suppose that B = {x₁, x2, ..., xn} and ß′ = {x'1, x′2, ……., x'n} are ordered bases for a vector space and Q is the change of coordinate matrix that changes B'-coordinates into B-coordinates. Then the jth column of Q is [xj] ¹² Every change of coordinate matrix is invertible. Let T be a linear operator on a finite-dimensional vector space V, let B and B' be ordered bases for V, and let Q be the change of coordinate matrix that changes 3-coordinates into B-coordinates. Then [T] = Q[T] „Q¯¹. В The matrices A, B = Mnxn (F) are called similar if B = Q AQ for some QE Mnxn (F). Let T be a linear operator on a finite-dimensional vector space V. Then for any ordered bases 3 and y for V, [T] is similar to [T] B 'y
This is a T/F question from the book. For all T/F, provide a brief justification for your answer. That may be citing an appropriate theorem or providing a counterexample.
Transcribed Image Text:This is a T/F question from the book. For all T/F, provide a brief justification for your answer. That may be citing an appropriate theorem or providing a counterexample.
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