# Give the matrix for T relative to the bases B and C

I can't figure out what I am doing wrong on this problem.
Let B be the basis of R2 given by $$\{\begin{bmatrix}2\\1\end{bmatrix},\begin{bmatrix}2\\-1\end{bmatrix}\}$$ Let C be the basis of R3 given by $$\{\begin{bmatrix}1\\2\\1\end{bmatrix},\begin{bmatrix}-2\\-2\\0\end{bmatrix},\begin{bmatrix}-1\\-1\\1\end{bmatrix}\}$$ Let T: R2 --> R3 be the transformation with standard martix $$\begin{bmatrix} 1 & -2 \\ 2 & 2 \\ 1 & 0 \\ \end{bmatrix}$$

I did: $$\begin{bmatrix} 1 & -2 & -1 & 2 & -1 \\ 2 & 2 & -1 & -1 & 4 \\ 1 & 0 & 1 & 2 & 1 \\ \end{bmatrix}$$

Row reduced and took the last two columns to get $$\begin{bmatrix} 1 & 1 \\ -1 & 1 \\ 1 & 0 \\ \end{bmatrix}$$

Did I not set this problem up correctly or something? No comments were made on this problem so I do not even have a clue.

Recall that the columns of a transformation matrix are the images of the basis vectors, so the matrix we want is $TB$ expressed relative to the $C$-basis. The matrix $C$ converts from the $C$-basis to the standard basis, so we left-multiply by its inverse to get $C^{-1}TB$ as our matrix.
To compute this, you can form the augmented matrix $\left[\begin{array}{c|c}C&TB\end{array}\right]$ and row-reduce it to get $\left[\begin{array}{c|c}I&C^{-1}TB\end{array}\right]$. Instead you’ve started with $\left[\begin{array}{c|c}C&T\end{array}\right]$ and ended up with $\left[\begin{array}{c|c}I&C^{-1}T\end{array}\right]$.