Linear Maps Determined by Their Actions on A Basis

Suppose $V$ is a finite dimensional vector space with basis $\alpha =\{v_1,\dots ,v_n\}$ If $S:V\to W$ and $T:V\to W$ are Linear Maps s.t $S(v_1)=T(v_1)$ For each $v_i$ in the basis $\alpha$, then $S(v)=T(v)$ for all $v\in V$

Intuition

Every transformed vector in a linear map is determined based off its basis vectors

Proof

Suppose $T(v_i)=S(v_i)$ for all $v_i\in \alpha$ Pick any vector $v\in V$ We can represent $v$ uniquely using the basis $\alpha$ as $v=\alpha v_1+\cdots +a_n{v}_n$ Where: $a_i\in F$ and $v_i\in \alpha$. This gives: $S(v)=S(a_1{v}_1+\cdots +a_n{v}_n)$ $=a_{1}S(v_1)+\cdots +a_{n}S(v_n)$ $=a_{1}T(v_1)+\cdots +a_{n}T(v_n)$ as $S(v_i)=T(v_i)$ $=T(a_1{v}_1+\cdots +a_n{v}_n)$ $=T(v)$ Therefor: $S(v)=T(v)$ for all $v\in V$ $\square$