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

  1. differentiable everywhere on $\mathbb{R}^2$

  2. differentiable on $(0,0)$ only

  3. $DL(0,0)=L$

  4. $ 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.

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Yes, you're correct. $L$ is a linear map so its derivative is itself. –  Christopher A. Wong May 22 '13 at 10:18
    
@ChristopherA.Wong Thank you very much –  El Angel Exterminador May 22 '13 at 10:21

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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.

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