Linear Transform: Proving Action Determined by Basis

[DanielFaraday]

Homework Statement

Prove the following:

The action of a linear transformation T:U\rightarrow V is completely determined by its action on a basis B=\left\{u_1,u_2,\text{...},u_n\right\} for the domain U.

Homework Equations

None

The Attempt at a Solution

Okay, I feel like my solution is too simple. It doesn't feel rigorous enough, but I'm not sure how to improve it. Any ideas?

By definition, the linear transformation T:U\rightarrow V only acts on the domain U. Also by definition, a basis of U must also span U and be linearly independent. Thus, U is completely determined by B, and in turn, T is completely determined by its action on B. QED

 

[CompuChip]

You haven't really proven it yet.
I'd start by: suppose that T(B) is given, i.e. T(u_i) = v_i. Let u \in U. You should now be able to write down T(u).