How to find $[T]_E$ when given $T:V \to V$ and $T = T^2$?

Question

How to find $[T]_E$ when given $T:V \to V$ and $T = T^2$ ? $E$ is the standard basis. I want to find $[T]$ because later I will use this to find the eigenvalues of $T$

Answer 1

If $T^2=T$ and $v$ is an eigenvector corresponding to the eigenvalue $\lambda$, then $\lambda v=T(v)=T^2(v)=\lambda^2v$. Hence $\lambda^2=\lambda$. It follows that $\lambda=0$ or $1$. As stated in your other question, you lack any information to find the matrix of $T$.

Answer 2

You cannot find $T$ or $[T]$ because it is not uniquely determined by the data given. For instance, for $B = \mathbb R$, $T$ could either be $T(x) = 0$ or $T(x) = x$.