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(2) If V and W are two finite-dimensional vector spaces, under which of the following circumstances are V and W not necessarily isomorphic? • There are one-to-one linear transformations T: V → W and S: W → V. • There is a one-to-one linear transformation T: V → W and an onto linear transformation S: V → W. • The dimension of the space L(V, W) of linear transformations is a perfect square.

(2) If V and W are two finite-dimensional vector spaces, under which of the following circumstances are V and W not necessarily isomorphic? • There are one-to-one linear transformations T: V → W and S: W → V. • There is a one-to-one linear transformation T: V → W and an onto linear transformation S: V → W. • The dimension of the space L(V, W) of linear transformations is a perfect square.

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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The following questions are all multiple-choice. In these questions, you must state which answer you believe to be correct, provide a justification/proof that your chosen answer is correct, and also provide a justification/proof that the other two answers are incorrect. (2) If V and W are two finite-dimensional vector spaces, under which of the following circumstances are V and W not necessarily isomorphic? • There are one-to-one linear transformations T: V → W and S : W → V. • There is a one-to-one linear transformation T: V → W and an onto linear transformation S : V → W. The dimension of the space L(V, W) of linear transformations is a perfect square. is a finite-dimensional inner product space and V = W1 W2 is the direct sum of two subspaces,
Transcribed Image Text:The following questions are all multiple-choice. In these questions, you must state which answer you believe to be correct, provide a justification/proof that your chosen answer is correct, and also provide a justification/proof that the other two answers are incorrect. (2) If V and W are two finite-dimensional vector spaces, under which of the following circumstances are V and W not necessarily isomorphic? • There are one-to-one linear transformations T: V → W and S : W → V. • There is a one-to-one linear transformation T: V → W and an onto linear transformation S : V → W. The dimension of the space L(V, W) of linear transformations is a perfect square. is a finite-dimensional inner product space and V = W1 W2 is the direct sum of two subspaces,
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