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Let F be a field of scalars, let V and W be vectorspaces over F, and let T: V → W be a linear transformation. Let S = {v1,..., Vk} CV be a set of vectors in V and let T(S) = {T(v;) | v; E S} C W. Vi

Let F be a field of scalars, let V and W be vectorspaces over F, and let T: V → W be a linear transformation. Let S = {v1,..., Vk} CV be a set of vectors in V and let T(S) = {T(v;) | v; E S} C W. Vi

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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Linear Transformation

Let F be a field of scalars, let V and W be vectorspaces over F, and let T: V → W be a linear transformation. Let S = {v1,….., vk} CV • • . be a set of vectors in V and let T(S) = {T(v;) | v; E S} C W. (a) Prove: If S is linearly dependent inside V, then T(S) is linearly dependent inside W. (b) Assume that S is an ordered basis of V (the ith element is v;) and that T(S) is an ordered basis of W (the ith element is T(v;)). Fix a vector v E V. Prove: The coordinate vector of v with respect to S is equal to the coordinate vector of T(v) with respect to T(S): [v]s = [T(v)]T(s).
Transcribed Image Text:Let F be a field of scalars, let V and W be vectorspaces over F, and let T: V → W be a linear transformation. Let S = {v1,….., vk} CV • • . be a set of vectors in V and let T(S) = {T(v;) | v; E S} C W. (a) Prove: If S is linearly dependent inside V, then T(S) is linearly dependent inside W. (b) Assume that S is an ordered basis of V (the ith element is v;) and that T(S) is an ordered basis of W (the ith element is T(v;)). Fix a vector v E V. Prove: The coordinate vector of v with respect to S is equal to the coordinate vector of T(v) with respect to T(S): [v]s = [T(v)]T(s).
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