$L:\mathbb{R}^2\to\mathbb{R}^2$ given by $L(x,y)=(x,-y)$ which of the following is true? $L:\mathbb{R}^2\to\mathbb{R}^2$ given by $L(x,y)=(x,-y)$ which of the following is true?


*

*differentiable everywhere on $\mathbb{R}^2$

*differentiable on $(0,0)$ only

*$DL(0,0)=L$

*$ DL(x,y)=L$ for all $(x,y)$
I have calculated that derivative matrix is $DL=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$
so $DL(x,y)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\times (x,y)^T=(x,-y)$ so $DL=L$ so $4$ is true right? and hence also $1$ is true.

This CW answer intends to remove the question from the unanswered queue.

As Christopher A. Wong already noted you have correctly calculated that 4 is true. And hence also 1 is true. Note that 4 being true also implies 3 being true.
