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(f) V = {x € R: In converges as noo}, W = R, T(x) = limn → In, I EV; (g) V =R%, W = R, T(x) = -1n, IEV;

(f) V = {x € R: In converges as noo}, W = R, T(x) = limn → In, I EV; (g) V =R%, W = R, T(x) = -1n, IEV;

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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part F G

3 Linear Mappings In which of the following cases is the mapping TV → W linear (F= R)? (a) * V = W = R², T(x, y) = (y,x), x, y ≤ R; (b) V = W = R³, T(x, y, z) = (y-2,2-x, x-y), x, y, z € R; (c) * V = P₂ (R), W = R², T(p) = (p(1), p(2)), p = P₂(R); (d) V = P₂ (R), W = R³, T(p) = (1,p(1), p(2)), p = P₂ (R); (e) V = P₂ (R), W = R, T(p) = (p(1), p(2), p(3),...), p = P₂ (R); (f) V = {x € R: In converges as n →∞}, W = R, T(x) = limnon, TV; (g) V =R%, W = R, T(x) = 1 In, I € V; n=1
Transcribed Image Text:3 Linear Mappings In which of the following cases is the mapping TV → W linear (F= R)? (a) * V = W = R², T(x, y) = (y,x), x, y ≤ R; (b) V = W = R³, T(x, y, z) = (y-2,2-x, x-y), x, y, z € R; (c) * V = P₂ (R), W = R², T(p) = (p(1), p(2)), p = P₂(R); (d) V = P₂ (R), W = R³, T(p) = (1,p(1), p(2)), p = P₂ (R); (e) V = P₂ (R), W = R, T(p) = (p(1), p(2), p(3),...), p = P₂ (R); (f) V = {x € R: In converges as n →∞}, W = R, T(x) = limnon, TV; (g) V =R%, W = R, T(x) = 1 In, I € V; n=1
In case that both V and W are finite-dimensional and T is a linear mapping, find the matrix representation M(T) of the mapping assuming canonical basis of V and W.¹ Re- member that the k-th column of this matrix represents T(uk), where u is the k-th element of the basis of V. Make sure that if dim(V) = n and dim(W) = m, then you should obtain m-by-n matrix.
Transcribed Image Text:In case that both V and W are finite-dimensional and T is a linear mapping, find the matrix representation M(T) of the mapping assuming canonical basis of V and W.¹ Re- member that the k-th column of this matrix represents T(uk), where u is the k-th element of the basis of V. Make sure that if dim(V) = n and dim(W) = m, then you should obtain m-by-n matrix.
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