Effects of Isomorphic Transformations on Vector Spaces.

Question

Let $V$, $W$ be finite-dimensional vector spaces and let $T: V\rightarrow W$ be an isomorphism.

Let $X$ be a subspace of $V$.

Show that $T(X)$ is a subspace of $V$.

My attempt:

I know two vector spaces are isomorphic if their dimensions are equal. I also know that an isomorphic transformation is one-to-one and onto. Also, because the transformation is isomorphic I know that the $\dim(T(X)) = \dim (X)$. It makes sense that if $X$ is a subspace of $V$, that $X$ is also a subspace of $W$ - because $V$ and $W$ are isomorphic. However, I don't know how to "mathematically" word this.

Any help would be appreciated.

Thank you.

Answer 1

In order to show that $T(X)$ is a subspace of W, you should show that $T(X)$ satisfies the definition of a subspace. That is:

• if $y \in T(X)$, then $ky \in T(X)$ for a scalar $k$
• if $y_1,y_2 \in T(X)$, then $y_1 + y_2\in T(X)$

Note that $y \in T(X)$ if and only if there exists an $x \in V$ such that $y = T(x)$.

As it turns out, the fact that $T$ is an isomorphism (as opposed to any other linear transformation) does not affect the proof.