Exponent of Projection (T^2=T)

Question

T:V->V is a linear transformation, also it's a projection, i.e. T^2=T.

Find e^T.

I thought of using the fact that if T=T^2 then e^T=e^(T^2) but I guess that doesn't work because exponent is a sum of endless "numbers" maybe the fact that T^k = T for every k may help ?

thanks

Answer 1

We have by a simple induction $T^n=T$ so

$$\exp (T)=\sum_{n=0}^\infty \frac{T^n}{n!}=I+T\sum_{n=1}^\infty\frac1{n!}=I+(e-1)T$$

Answer 2

Use the series representation of the exponential $$ e^T = \sum_{n=0}^\infty \frac1{n!} T^n, $$ from $T=T^2$ you can deduce $T^n=T$ for all $n$.