I have encountered a problem related to change of basis (not homework, though school-related) that I need some help understanding.
Suppose you have some basis for an old vector space
$$O = \begin{bmatrix}4 & 2 \\ -1 & 2\end{bmatrix}$$
and you receive a new basis
$$N = \begin{bmatrix}6 & 0 \\ 1 & 2\end{bmatrix}$$
Find the Transformation matrix $T$ so that
$$[x]N = T[x]B$$
given any vector $x$.
At first I thought this was a simple matter of just transforming vector space from $B$ to the standard coordinate space and then applying $N$ to that transformation. So I computed the inverse of $B$ and multiplied it with $N$. However when I checked the correct solution, which is given as
$$T= 1/6 *\begin{bmatrix}4 & 2 \\ -5 & 5\end{bmatrix}$$
it does not match my result at all. What am I missing and/or doing wrong?
Best Answer
For $x=\pmatrix{4\\-1}$, we have $[x]_B=\pmatrix{1\\0}$, so $$T\pmatrix{1\\0}=[x]_N$$ This will give the first column.
Similarly, by calculating $[x]_N$ for $x=\pmatrix{2\\2}$ we get the second column of $T$.
Note that $N\cdot [x]_N=x$, i.e. $[x]_N=N^{-1}x$, so we arrive to $T=N^{-1}B$, and thus it seems you basically inverted the wrong matrix.