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