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Let V and W be finite-dimensional vector spaces and T: V > W be an isomorphism. Let Vn

Chapter 2, Problem 17

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QUESTION:

Let V and W be finite-dimensional vector spaces and T: V > W be an isomorphism. Let Vn be a subspace of V. (a) Prove that T(V0) is a subspace of W. (b) Prove that dim(V0) = dim(T(V0)

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QUESTION:

Let V and W be finite-dimensional vector spaces and T: V > W be an isomorphism. Let Vn be a subspace of V. (a) Prove that T(V0) is a subspace of W. (b) Prove that dim(V0) = dim(T(V0)

ANSWER:

        

Step 1 of 2

Since is an isomorphism, it is one-to –one and onto. Since  is a vector subspace of is a vector subspace of W. Moreover both  and  must be finite dimensional. Now consider the restricted mapping

        

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