Answer to Question #151364 in Linear Algebra for Ashweta Padhan

Question

Give an example of a non linear transformation T:R^2 into R^2 such that
T^-1 (0)=0 but T is not one - one.

Answer 1

Let $T:\mathbb{R}^2\to \mathbb{R}^2$ such that $T(x_1,x_2)=(2x_2-2x_1,x_1-x_2).$

We can see that;

$T(0,0)=(2.0-2.0,0-0)\\ \implies T(0,0)=(0,0)\\ \text{pre-multiply } T^{-1} \text{ to both sides}\\ \implies (0,0)=T^{-1}(0,0)\\ \implies T^{-1}(0)=0$

Also, we can see that;

$Ker(T):=\{(x_1,x_2)|x_1=x_2\}$

And, the basis of $Ker(T)=\{(1,1)\}\implies dim(Ker(T))=1$

Hence, $T$ is not one-to-one