Finding matrix with respect to given bases Given that A:
\begin{matrix}
  a & b & c \\
  d & e & f \\
 \end{matrix}
is a matrix of T : V -> W with bases G = {g1, g2, g3} and Q = {q1, q2}, respectively.  Find the matrix of T with respect to:
(a) H = {g3, g2, g1} and R = {q1, q2}
(b)H = {g1, g2, g3} and R = {q2, q1}
 A: Definition:
$$A=\pmatrix{a&b&c\cr d&e&f\cr}$$
is the matrix of $T:V\to W$ with respect to bases $G=\{{\bf g}_1,{\bf g}_2,{\bf g}_3\}$ for $V$ and $Q=\{{\bf q}_1,{\bf q}_2\}$ for $W$ means that
$$[T({\bf v})]_Q=A\,[{\bf v}]_G$$
for all ${\bf v}\in V$, where square brackets denote coordinate vectors with respect to the specified basis.
Hint.  Take ${\bf v}={\bf g}_1$.  Then the definition says
$$[T({\bf g}_1)]_Q=A\pmatrix{1\cr0\cr0\cr}=\pmatrix{a\cr d\cr}\ ,$$
in other words,
$$T({\bf g}_1)=a{\bf q}_1+d{\bf q}_2\ .$$
For part (b), where ${\bf q}_1$ and ${\bf q}_2$ are given in the reverse order, this last step will need to be
$$T({\bf g}_1)=?{\bf q}_2+?{\bf q}_1\ .$$
So, what are the scalars which must replace $a,d$ (indicated by question marks in the last equation)?
A: Let $P$ the matrix change from the basis $G$ to the basis $H$ so we have
$$P=\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}$$
and $C$ the matrix change from the basis $R$ to the basis $Q$ then in your case $(a)$ we have
$$C=I_2$$
so the matrix of $T$ with respect to $H$ and $R$ is
$$I_2AP=AP$$
and notice that $AP$ can be obtained immediately by interchanging the columns of $A$ with respect to the permutation $(1\;3)$.
Can you now answer the case $(b)$?
