Is the inverse of a linear transformation linear as well?

Question

A question from the field of Linear Algebra. If I have a linear transformation $T$ that is one-to-one and onto, would that mean that the $T^{-1}$ will also be linear? If so, is there any general proof for it?

Thanks

Answer 1

Hint: Write down the equations saying that $T$ is linear. Now replace each variable $x$ with $T^{-1}(x)$ throughout, and apply $T^{-1}$ to both sides of the equation. Simplify.

Answer 2

Use the relation $$T^{-1} \circ T = {\rm Id}$$ and linearity of $T$ and $\rm Id$ to obtain $$T^{-1} (a T(v) + b T(w)) = av + bw.$$ Now write $v' = T(v)$ and $w' = T(w)$. We get $$T^{-1} (a v' + b w') = aT^{-1}(v') + b T^{-1}(w').$$