--- $\textbf{Exercise 30 Commentary:}$ This exercise shows that if $\phi \in \mathcal{L}(V, \mathbb{F})$ is a non-zero linear functional on a finite-dimensional vector space $V$, and $u \in V$ is not in the null space of $\phi$, then $V$ can be decomposed as a direct sum of the null space of $\phi$ and the one-dimensional subspace spanned by $u$. The proof consists of two main steps: 1) Showing that $\text{null}(\phi) \cap \text{span}\{u\} = \{0\}$. This follows from the fact that if $au \in \text{null}(\phi)$, then $\phi(au) = a\phi(u) = 0$, which implies $a = 0$ since $\phi(u) \neq 0$ by assumption. 2) Showing that every $v \in V$ can be written as $v = (v - \frac{\phi(v)}{\phi(u)}u) + \frac{\phi(v)}{\phi(u)}u$, where the first term is in $\text{null}(\phi)$ and the second term is in $\text{span}\{u\}$. Combining these two steps and applying the direct sum criterion (Theorem 1.46), we conclude that $V = \text{null}(\phi) \oplus \text{span}\{u\}$. $\textbf{Exercise 30 Examples:}$ 1) Let $V = \mathbb{R}^3$, and define $\phi: V \to \mathbb{R}$ by $\phi(x,y,z) = x+2y-z$. Take $u = (1,1,1)$. Then $\text{null}(\phi) = \text{span}\{(2,1,1), (-1,0,1)\}$, and $V = \text{null}(\phi) \oplus \text{span}\{(1,1,1)\}$. 2) Let $V = \mathcal{P}_2(\mathbb{R})$, and define $\phi: V \to \mathbb{R}$ by $\phi(a+bx+cx^2) = a$. Take $u = 1$. Then $\text{null}(\phi) = \text{span}\{x, x^2\}$, and $V = \text{null}(\phi) \oplus \text{span}\{1\}$. 3) Let $V = \mathbb{F}_4^2$, and define $\phi: V \to \mathbb{F}_4$ by $\phi(x,y) = x+2y$. Take $u = (1,1)$. Then $\text{null}(\phi) = \text{span}\{(2,1)\}$, and $V = \text{null}(\phi) \oplus \text{span}\{(1,1)\}$. $\textbf{Implications and Applications:}$ This result has important implications in the study of linear functionals and their matrix representations, as well as in the development of efficient algorithms for computing null spaces and ranges of matrices. Linear functionals play a crucial role in the theory of duality and the study of dual spaces, which are fundamental concepts in linear algebra and functional analysis. The direct sum decomposition provided by this exercise allows for a deeper understanding of the algebraic and geometric properties of linear functionals, 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.