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Let V and W be n-dimensional vector spaces, and let T: V * W be a linear transformation

Chapter 2, Problem 15

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QUESTION:

Let V and W be n-dimensional vector spaces, and let T: V * W be a linear transformation. Suppose that 0 is a basis for V. Prove that T is an isomorphism if and only if T(0) is a basis for W.

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QUESTION:

Let V and W be n-dimensional vector spaces, and let T: V * W be a linear transformation. Suppose that 0 is a basis for V. Prove that T is an isomorphism if and only if T(0) is a basis for W.

ANSWER:

Step 1 of 3

Assume that be an isomorphism, where and is an n-dimensional vector space.

Let be a basis for .

Then,

Let

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