--- $\textbf{Exercise 27 Commentary:}$ This exercise shows that if $P \in \mathcal{L}(V)$ is a linear operator on a finite-dimensional vector space $V$ that satisfies $P^2 = P$, then $V$ can be decomposed as a direct sum of the null space of $P$ and the range of $P$. The proof consists of two main steps: 1) Showing that $\text{null}(P) \cap \text{range}(P) = \{0\}$. This follows from the fact that if $Pv = v \neq 0$, then applying $P$ again yields $P^2v = Pv \neq 0$, contradicting the assumption that $P^2 = P$. 2) Showing that $V = \text{null}(P) + \text{range}(P)$. For any $v \in V$, we can write $v = (v - Pv) + Pv$, where $v - Pv \in \text{null}(P)$ and $Pv \in \text{range}(P)$. Combining these two steps and applying the direct sum criterion (Theorem 1.46), we conclude that $V = \text{null}(P) \oplus \text{range}(P)$. $\textbf{Exercise 27 Examples:}$ 1) Let $V = \mathbb{R}^3$, and define $P: V \to V$ by $P(x,y,z) = (x,0,0)$. Then $P^2 = P$, and we have $\text{null}(P) = \text{span}\{(0,1,0), (0,0,1)\}$, $\text{range}(P) = \text{span}\{(1,0,0)\}$, and $V = \text{null}(P) \oplus \text{range}(P)$. 2) Let $V = \mathcal{P}_3(\mathbb{R})$, and define $P: V \to V$ by $P(a + bx + cx^2 + dx^3) = a + bx$. Then $P^2 = P$, and we have $\text{null}(P) = \text{span}\{1, x^2, x^3\}$, $\text{range}(P) = \mathcal{P}_1(\mathbb{R})$, and $V = \text{null}(P) \oplus \text{range}(P)$. 3) Let $V = \mathbb{F}_2^4$, and define $P: V \to V$ by $P(x_1, x_2, x_3, x_4) = (x_1, x_2, 0, 0)$. Then $P^2 = P$, and we have $\text{null}(P) = \text{span}\{(0,0,1,0), (0,0,0,1)\}$, $\text{range}(P) = \text{span}\{(1,0,0,0), (0,1,0,0)\}$, and $V = \text{null}(P) \oplus \text{range}(P)$. $\textbf{Implications and Applications:}$ This result has important implications in the study of linear operators and their matrix representations, as well as in the development of efficient algorithms for computing null spaces and ranges of matrices. Linear operators satisfying $P^2 = P$ are called projection operators, and they play a crucial role in the study of linear transformations and their compositions. The direct sum decomposition provided by this exercise allows for a deeper understanding of the algebraic and geometric properties of these operators, as well as their behavior under compositions and other operations. The result also has applications in the study of matrix decompositions and canonical forms, as well as in the analysis of linear systems with singular or ill-conditioned coefficient matrices. It connects to the theory of generalized inverses and the study of linear systems with singular or ill-conditioned coefficient matrices.