Transformation Defined by Matrix

• Fix bases $\alpha =\{v_1,\dots ,v_n\}$ of $V$
• $\beta =\{w_1,\dots ,w_k\}$ of $W$
• If $M\in M_{k\times n}(\mathbb{R})$
• Then, the transformation $M$ is $T_M:V\to W$ s.t $T_M(v)=w⟺M[v{]}_{\alpha }=[w{]}_{\beta }$