Show a subset A is compact if and only if the image of the map T(A) is compact

Question

The question is as follows:

Let $\{v_1,v_2,\dots,v_n\}$ be a set of linearly independent vectors of an $n$-dimensional normed linear vector space $V$. Define the map $T: R^n \to V$ by $T(a) = \Sigma_{i=1}^na_iv_i$ for any $a\in R^n$. Let A be a subset of Rⁿ.

Then, I want to show that A is compact if and only if T(A) is compact.

I was able to show that A is bounded if and only if T(A) is bounded, and I was thinking about next showing that A is closed and equicontinuous iff T(A) is closed and equicontinuous (then I could apply the Arzela Ascoli theorem) but I was wondering if there was a more direct way to show compactness.

Any help would be appreciated, thanks.