Verification of change of basis calculation A short question just to check regarding a change of basis: Let $$A = \left\{
  \begin{bmatrix}
    2  \\
    1 
  \end{bmatrix}, \begin{bmatrix}
    2  \\
    5 
  \end{bmatrix} \right\}$$
be a basis
and let $A'$ be the standard basis of $\mathbb{R}^2$, would I be right in stating that the change of basis matrix is $P = \begin{bmatrix}
    2 & 2  \\
    1 & 5
  \end{bmatrix}$ then $P^{-1} = \begin{bmatrix}
    \frac{5}{8} & -\frac{1}{4}  \\
    -\frac{1}{8} & \frac{1}{4}
  \end{bmatrix}$ and that $C =\begin{bmatrix}
    -1  \\
    2 
  \end{bmatrix}$ written in the standard basis is $$P^{-1}C = \begin{bmatrix}
    -\frac{9}{8}   \\
    \frac{5}{8}
  \end{bmatrix} 
.$$
Is this the correct working, for some reason it's not agreeing with the answer in the literature which gives \begin{bmatrix}
    2   \\
    9 
  \end{bmatrix}
Thanks.
 A: You should be careful with "the" matrix for a change of basis, since the order matters: the matrix converting coordinates from basis $A$ to basis $A'$ is the inverse of the matrix converting coordinates in the other direction.
You filled a matrix with as columns the base vectors of $A$ expressed w.r.t. the base vectors of $A'$, the standard basis. This gives you a change of basis matrix $P_{A'A}$ that converts coordinates in the following direction:
$$[\vec x]_{A'} = P_{A'A}[\vec x]_{A}$$
where $[\vec x]_B$ denotes the coordinate vector of $\vec x$ w.r.t. a basis $B$.
You use it the other way around, there was no need to find the inverse matrix for the conversion in the direction asked.
If a vector $\vec c$ has coordinate vector:
$$[\vec c]_{A} = \begin{bmatrix}
    -1   \\
    2 
  \end{bmatrix}$$
with respect to $A$, then the coordinate vector with respect to $A'$ is given by:
$$[\vec c]_{A'} = P_{A'A}[\vec c]_{A} = \begin{bmatrix}
    2 & 2  \\
    1 & 5
  \end{bmatrix}\begin{bmatrix}
    -1   \\
    2 
  \end{bmatrix}= \begin{bmatrix}
    2   \\
    9 
  \end{bmatrix} $$
If you are given a coordinate vector w.r.t. the standard basis and you would want to know its coordinate vector w.r.t. the intially given basis $A$, then you would need $P_{AA'} = P_{A'A}^{-1}$.
