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 $\displaystyle A:R^2\to R^2$ linear transformation which is Bijection. Let us show now that $\displaystyle A^{-1}:R^2\to R^2$ is also linear transformation.
Suppose $\displaystyle a,b \in R^2$, $\displaystyle A$ is a bijection, there are exist unique vectors $\displaystyle c,d \in R^2$ which for them:
$\displaystyle A(c)=a$ and $\displaystyle A(d)=b$.
From the linearity of $\displaystyle A$ we have also:
$\displaystyle A(c+d)=A(c)+A(d)=a+b$ and $\displaystyle A(kc)=kA(c)=ka$.
Now, by definition of inverse transformation, $\displaystyle A^{-1}(a)=c$ , $\displaystyle A^{-1}(b)=d$ , $\displaystyle A^{-1}(a+b)=c+d$ and $\displaystyle A^{-1}(ka)=kc$.
Hence:
$\displaystyle A^{-1}(a+b)=c+d=A^{-1}(a)+A^{-1}(b)$ and $\displaystyle A^{-1}(ka)=kc=A^{-1}(a)$, therefor $\displaystyle A$ is linear transformation.