I'm not sure if I'm solving the following problem correctly so I would really appreciate if someone could help me out here.
Given a matrix $A = \begin{bmatrix}-37 & 24\\-60 & 39\end{bmatrix}$,
and $B = \{[2,3] , [3,5]\}$ is a basis of $\mathbb{R}^2$ consisting of eigenvectors for $A$.
I need to find the change of coordinate matrix $P = _S P_B$ where $S$ is the standard basis.
This is what I'm thinking I'm supposed to do but I'm really unsure. Can anyone let me know if I'm on the right track?
$\begin{bmatrix}1 & 0\end{bmatrix}$ $x_1$ + $\begin{bmatrix}0 & 1\end{bmatrix}$ $x_2$ = $\begin{bmatrix}-37 & 24\\-60 & 39\end{bmatrix}$
Best Answer
There's, I'll adjust what @Bernard said slightly, almost no computation here. Well, it depends on whether you know an all-important secret or two.
There's the transition matrix, and on the other hand the change of basis matrix. They're inverses. (I tend to forget which is which).
But the matrix in this case is $$\begin{pmatrix}2\quad3\\3\quad 5\end{pmatrix}$$.
The reason is that it takes vectors expressed in terms of the basis consisting of the columns, and expresses them in the standard basis. This can be easily checked, by applying the matrix to the elements of the basis consisting in the columns, expressed in terms of itself. That means applying it to $\begin {pmatrix}1\\0\end {pmatrix}$ and $\begin{pmatrix}0\\1\end{pmatrix}$. And as you see out pop the columns, in the standard basis.
Now, back to what @Bernard said. Often one will need to go in the other direction. Thus you do need to compute the inverse of this matrix.