## Idempotent Linear Transformation and Direct Sum of Image and Kernel

## Problem 327

Let $A$ be the matrix for a linear transformation $T:\R^n \to \R^n$ with respect to the standard basis of $\R^n$.

We assume that $A$ is idempotent, that is, $A^2=A$.

Then prove that

\[\R^n=\im(T) \oplus \ker(T).\]