Question

Considering (V) and (W) as (n)-dimensional vector spaces and (T: V rightarrow W) as a linear transformation, if (B) is a basis for (V), prove (T) is an isomorphism if and only if (TB) is a basis for (W).

A. (T) is an isomorphism if (TB) spans (W)
B. (T) is an isomorphism if (TB) is linearly independent
C. (T) is an isomorphism if (TB) is not invertible
D. (T) is an isomorphism if (TB) forms a subspace of (W)

Answer 1

To show T is an isomorphism if and only if TB is a basis for W, one must prove that if T is an isomorphism, then TB is a basis for W, and conversely, if TB is a basis for W, then T is an isomorphism.

Step-by-step explanation:

To prove that a linear transformation T from an n-dimensional vector space V to an n-dimensional vector space W is an isomorphism if and only if TB (the image of basis B of V under T) is a basis for W, we need to show two implications:

• If T is an isomorphism, then TB is a basis for W.
• If TB is a basis for W, then T is an isomorphism.

Proof:

1. Assume T is an isomorphism: This implies T is bijective (one-to-one and onto). Because T is one-to-one, the images of the basis vectors in B are linearly independent in W. Since T is also onto, the images of basis vectors span W. Hence, TB is a basis for W.
2. Assume TB is a basis for W: This implies that TB is linearly independent and spans W. Since TB spans W, T is onto. Because TB is linearly independent, T is one-to-one. Therefore, T is an isomorphism.

The option that is correct is B: 'T is an isomorphism if TB is linearly independent.'