prove inverse transformation is linear

Could someone help me with the following problem?

Suppose A is a linear transformation from the x-y plane to itself. Show that A^-1 is also a linear transformation (if it exists).

Thanks for the help!

Quote:

Originally Posted by sdh2106

Could someone help me with the following problem?

Suppose A is a linear transformation from the x-y plane to itself. Show that A^-1 is also a linear transformation (if it exists).

Thanks for the help!

Suppose $A:R^2\to R^2$ linear transformation which is Bijection. Let us show now that $A^{-1}:R^2\to R^2$ is also linear transformation.

Suppose $a,b \in R^2$ , $A$ is a bijection, there are exist unique vectors $c,d \in R^2$ which for them:

$A(c)=a$ and $A(d)=b$ .

From the linearity of $A$ we have also:

$A(c+d)=A(c)+A(d)=a+b$ and $A(kc)=kA(c)=ka$ .

Now, by definition of inverse transformation, $A^{-1}(a)=c$ , $A^{-1}(b)=d$ , $A^{-1}(a+b)=c+d$ and $A^{-1}(ka)=kc$ .

Hence:

$A^{-1}(a+b)=c+d=A^{-1}(a)+A^{-1}(b)$ and $A^{-1}(ka)=kc=A^{-1}(a)$ , therefor $A$ is linear transformation.

great explanation. thanks again!