Question

Prove that the inverse T^-1 of an invertible transformation T is linear, taking the definition of linear as your starting point.

Answer 1

A bit late to reply, but if in case you check back later:

To prove T^-1 is linear, you need to show two things:

T^-1(w+z) = T^-1(w) + T^-1(z) for all w,z in the range.
T^-1(c*z) = c*T^-1(z) for all y in the range, and where c is a scalar.

So, the trick is to write everything in terms of T, since all of x in T will map to something in T^-1, which will be all of T^-1. So, I'll show one of the two properties. The other one will follow almost exactly the same way:

Let x,y be in your domain. Since T is linear, T(x+y) = T(x) + T(y)
T(x) = w and T(y) = z for some w,z in the range.
So,

T^-1(w) + T^-1(z)
= x + y
= T^-1(T(x+y)) (since T^-1T is just the identity function)
= T^-1(T(x) + T(y)) since T is linear
= T^-1(w + z).

And, from there, additivity is proved. The second property uses the exact same trick.