Question: Let

\[S=\left\{ \left[ \begin{matrix} 1 \\ 2 \\ \end{matrix} \right],\left[ \begin{matrix} 0 \\ 1 \\ \end{matrix} \right] \right\},\,\,\,\,T=\left\{ \left[ \begin{matrix} 1 \\ 1 \\ \end{matrix} \right],\left[ \begin{matrix} 2 \\ 3 \\ \end{matrix} \right] \right\}\]

be ordered bases for \({{R}^{2}}\). Let

\[v=\left[ \begin{matrix} 1 \\ 5 \\ \end{matrix} \right],\,\,\,w=\left[ \begin{matrix} 5 \\ 4 \\ \end{matrix} \right]\]

(a) Find the coordinate vectors of v and w with respect to the basis T.

(b) What is the transition matrix \({{P}_{S\leftarrow T}}\) from the T-basis to the S-basis?

(c) Find the coordinate vectors of v and w with respect to S using \({{P}_{S\leftarrow T}}\)

(d) Find the coordinate vector of v and w with respect to S directly

(e) Find the transition matrix \({{Q}_{T\leftarrow S}}\) from the S-basis to the T-basis?

(f) Find the coordinate vectors of v and w with respect to T using \({{Q}_{T\leftarrow S}}\).