Please help with this Linear Algebra question which involves

Please help with this Linear Algebra question which involves 2 proofs. Show the transformation equals T^2 and that the null and range space equals the zero vector.

Thank you!

Solution

Note we always have ker(T) ker(T² ) ; Given, nullity(T) = nullity(T ² )

In other words, ker(T) and ker(T² ) have the same dimension.

But since ker(T) is a subspace of ker(T² ) of the same dimension, it must be equal to it.

So ker(T) = ker(T² )

=> ker(T ²) ker(T)

=> if T(T()) = 0, then T() = 0. --- (i)

We want to show ker(T) im(T) = {0}.

Of course, {0} ker(T) im(T), since kernel and image contain 0.

So it suffices to show ker(T) im(T) {0}.

So let v ker(T) im(T). We want to show v = 0.

since v im(T), we can write v = T(w) for some w V

On the other hand, since v ker(T), we see that T(v) = 0 i.e. T(T(w)) = 0

From (i),   T(w) = 0

So v = T(w) = 0, as desired.