I have an exam tomorrow and trying to understand this question:
Let $f(x,y)=(xy,y/x)$ compute $Df$. Compute the matrix of $Df(x,y)$ with respect to basis $(1,0),(1,1) $ in $ R^2$
First I found the matrix $Df = \begin{bmatrix}y & x\\-y/x^2 & 1/x\end{bmatrix}.$
Not sure what the next step is - I thought maybe:
$\begin{bmatrix}y & x\\-y/x^2 & 1/x\end{bmatrix}\begin{bmatrix}1 \\0\end{bmatrix} =\begin{bmatrix}y \\-y/x^2 \end{bmatrix} $
$\begin{bmatrix}y & x\\-y/x^2 & 1/x\end{bmatrix}\begin{bmatrix}1 \\1\end{bmatrix} =\begin{bmatrix}y+x \\-y/x^2 + 1/x\end{bmatrix} $
So the final answer: $\begin{bmatrix}y & y+x\\-y/x^2 & -y/x^2 + 1/x\end{bmatrix}$