Proof that it is sufficient to define T on basis vectors to determine the whole transformation?

Question

I want to make sure that I am understanding it correctly to only deal with basis vectors when I want to find a linear transformation.

Let ${v_i ... v_n}$ be a basis of V

Thus, for any $x\in V$, $x = \sum c_i*v_i$. This means that $T(x) = \sum c_i*T(v_i)$ meaning that knowing only what T does to the basis vectors is sufficient. Is this a rigorous proof?

Answer 1

Yes it is absolutely correct, once we know $T(v_i)$ we know everything since $\forall x$ we have $x=\sum a_iv_i$.