23-02-2025

Problem: Suppose $A$ is an $n \times n$ matrix with the property that $A^2 = A$. Let $\mathbf{C} (A)$ denotes the column space of $A$ and $\mathbf{N} (A)$ denotes the null space of $A$.
  1. Prove that $\mathbf{C} (A) = \left\{ \mathbf{x} \in \mathbb{R} ^n: \mathbf{x} = A \mathbf{x} \right\} $.
  2. Prove that $\mathbf{N} (A) = \left\{ \mathbf{x} \in \mathbb{R} ^n: \mathbf{x} = \mathbf{u} - A \mathbf{u} \text{ for some } \mathbf{u} \in \mathbb{R} ^n \right\} $.
  3. Prove that $\mathbf{C} (A) \cap \mathbf{N} (A) = \{ \mathbf{0} \} $.
  4. Prove that $\mathbf{C} (A)+ \mathbf{N} (A) = \mathbb{R} ^n$.