What will be the derivative of linear Transformation $L(x,y)=(x,-y)$

Question

Let $L$ be a map such that $L: \mathbb{R^2}\to \mathbb{R^2}$ given by $$L(x,y)=(x,-y)$$ In this question it is given that

$DL(0,0)$=$L(x,y)$

i am not getting this how this is true?.

Solution i tried-Given linear transformation is $L(x,y)=(x,-y)$,so its derivative at point $(x,y)$ will be $$ \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}=(x,-y) $$ which is again the same transformation . But as per option the point is $(0,0)$ so the derivative will be $0$.Does the option is wrong ?insted of $L(x,y)$ there is $L(0,0)$

$dL(0,0)$ is a linear map from $\mathbb R^{2}$ into itself. The question was really supposed to say $L(0,0)(x,y)=L(x,y)$. — Kavi Rama Murthy, Sep 16 '19 at 09:19

score 2 · Answer 1 · answered Sep 16 '19 at 09:17

2

Just apply the definition of derivative. The derivative of a linear map at any point is exactly the linear map itself.

answered Sep 16 '19 at 09:17

Kavi Rama Murthy

311,013

1

Yes, there is a mistype in the question. – Kavi Rama Murthy Sep 16 '19 at 09:21
Thankyou sir,I was thinking about this. – gaurav saini Sep 16 '19 at 09:24

What will be the derivative of linear Transformation $L(x,y)=(x,-y)$

1 Answers1