Changing Coordinates and Linear Maps

Theorem

• If $T:V\to W$ is a linear map
• With $[T{]}_{\alpha }^{\beta }$ as the matrix representation for $T$
• If $V$ has bases $\alpha ,{\alpha }^{\prime }$
• If $W$ has bases $\beta ,{\beta }^{\prime }$
• Then, $[T{]}_{{\alpha }^{\prime }}^{{\beta }^{\prime }}=[I_W{]}_{\beta }^{{\beta }^{\prime }}[T{]}_{\alpha }^{\beta }[I{]}_{{\alpha }^{\prime }}^{\alpha }$ Notice the composition $T\stackrel{I_V}{\to }V\stackrel{I_W}{\to }W$

Proof

• For every vector $v\in V$, we have $T(v)=I_W(T(I_V(v)))$
• Note that $[T{]}_{{\alpha }^{\prime }}^{{\beta }^{\prime }}=[I_{W}T{I}_V{]}_{{\alpha }^{\prime }}^{{\beta }^{\prime }}$
• $=[I_W{]}_{\beta }^{{\beta }^{\prime }}[T{]}_{\alpha }^{\beta }[I_V{]}_{{\alpha }^{\prime }}^{\alpha }$ by Matrix Vector Product and Matrix Multiplication of Composite Maps