Linear maps and eigenvectors

Question

Hello, I'm struggling to get anywhere with this question.

I know a map is linear if

$f(x+y) = f(x) + f(y)$

and

$f(ax)=a*f(x)$

I am also fairly familiar with eigenvalues for 2x2 matrices, but I'm not well adverse with general eigenvalues and eigenvectors.

Thanks

Answer 1

For a): try first to write $T^2v$. It is $T(Tv)=T(\lambda v)$, and since $T$ is linear...

Then try with $T^3, T^4$, and you'll soon find a general rule.

For b), remember that the inverse means $T^{-1}T=I$ (identity). Now, $Iv=v$, and $T^{-1}(Tv)=$...

Try yourself!