Find the transformation matrix of a linear mapping given 2 basis and only one image of one vector of each basis

Question

Linear mapping $g: R² ↦ R²$. They give me one basis: $B = {F_1 = (1,1), F_2 = (-1,1)}$ and another $C = {e_1 = (1,0), e_2 = (0,1)}$. In addition $g(e_2) = F_1$ and $g(F_2) = e_2$. I don't know how to get the form of the image, something like $g(x, y) = (2x, y-x)$ (with the correct values).

Sorry for my poor format i'm still novice. Thanks in advance!

Answer 1

Let the matrix of $g:\Bbb R^2\to\Bbb R^2$ be $\begin{bmatrix}a&b\\c&d\end{bmatrix}$.$$g(0,1)=\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}0\\1\end{bmatrix}=\begin{bmatrix}b\\d\end{bmatrix}=\begin{bmatrix}1\\1\end{bmatrix}\\g(-1,1)=\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}-1\\1\end{bmatrix}=\begin{bmatrix}b\\d\end{bmatrix}-\begin{bmatrix}a\\c\end{bmatrix}=\begin{bmatrix}0\\1\end{bmatrix}\\\implies\begin{bmatrix}a\\c\end{bmatrix}=\begin{bmatrix}1\\0\end{bmatrix}$$

This gives the matrix of $g$ as $\begin{bmatrix}1&1\\0&1\end{bmatrix}$.$$\implies g(x,y)=\begin{bmatrix}1&1\\0&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=(x+y,y)$$

Answer 2

$g(0,1)=(1,1)$
$ g(-1,1)=-g(1,0)+g(0,1)=(0,1)$
$-g(1,0) = (-1,0) \rightarrow g(e_1)=e_1$
$g(e_2)=e_1 + e_2$

So the matrix of the transformation in standard basis looks like this: \begin{bmatrix} 1 & 1\\ 0 & 1\\ \end{bmatrix}