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In each case below, V and W are vector spaces over a field F. Is V isomorphic to W? If yes, write down an isomorphism f: V→ W and explain why it is an isomorphism. If no, explain why no isomorphism can exist. (a) V = R¹ and W = C², where F = R. (b) V = Fun (S, Q²) and W=M3,4(Q), where S = {1, 2, 3, 4, 5, 6} and F = Q. (c) V = M₂,3(R) and W = C°(R, R), where and F = R. (d) V = R4 M3,3(R) and W = R", where F = R and n ≥ 1.

In each case below, V and W are vector spaces over a field F. Is V isomorphic to W? If yes, write down an isomorphism f: V→ W and explain why it is an isomorphism. If no, explain why no isomorphism can exist. (a) V = R¹ and W = C², where F = R. (b) V = Fun (S, Q²) and W=M3,4(Q), where S = {1, 2, 3, 4, 5, 6} and F = Q. (c) V = M₂,3(R) and W = C°(R, R), where and F = R. (d) V = R4 M3,3(R) and W = R", where F = R and n ≥ 1.

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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In each case below, V and W are vector spaces over a field F. Is V isomorphic to W? If yes, write down an isomorphism f: V→ W and explain why it is an isomorphism. If no, explain why no isomorphism can exist. (a) V = R¹ and W = C², where F = R. (b) V = Fun (S, Q²) and W=M3,4(Q), where S = {1, 2, 3, 4, 5, 6} and F = Q. (c) V = M₂,3(R) and W = C°(R, R), where and F = R. (d) V = R4 M3,3(R) and W = R", where F = R and n ≥ 1.
Transcribed Image Text:In each case below, V and W are vector spaces over a field F. Is V isomorphic to W? If yes, write down an isomorphism f: V→ W and explain why it is an isomorphism. If no, explain why no isomorphism can exist. (a) V = R¹ and W = C², where F = R. (b) V = Fun (S, Q²) and W=M3,4(Q), where S = {1, 2, 3, 4, 5, 6} and F = Q. (c) V = M₂,3(R) and W = C°(R, R), where and F = R. (d) V = R4 M3,3(R) and W = R", where F = R and n ≥ 1.
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