--- $\textbf{Exercise 19 Commentary:}$ This exercise provides a characterization of injective linear maps in terms of the existence of a "right inverse" map. Specifically, it shows that a linear map $T: V \to W$ is injective if and only if there exists a linear map $S: W \to V$ such that $ST$ is the identity map on $V$. The proof relies on the fact that injectivity of $T$ is equivalent to $\text{null}(T) = \{0\}$. If $T$ is injective, the proof constructs a right inverse $S$ by first defining it on the range of $T$ as $S(Tv) = v$, and then extending it to a linear map on all of $W$ using a result from an earlier exercise. Conversely, if $ST$ is the identity on $V$, then $T$ must be injective, as $Tu = Tv$ implies $u = (ST)u = (ST)v = v$. $\textbf{Exercise 19 Examples:}$ 1) Let $T: \mathbb{R}^2 \to \mathbb{R}^3$ be given by $T(x, y) = (x, 2y, 3x-y)$. Then $T$ is injective, and a right inverse is $S(a, b, c) = (a, (c+b)/3)$. 2) Let $V = \mathcal{P}_2(\mathbb{R})$ and $W = \mathbb{R}^3$, and define $T: V \to W$ by $T(a + bx + cx^2) = (a, b, c)$. Then $T$ is injective, with right inverse $S(x, y, z) = x + yz + z^2$. 3) Let $V = \mathbb{C}^4$ and $W = \mathbb{C}^3$, and let $T: V \to W$ be the linear map with matrix $\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix}$. Then $T$ is not injective, and there does not exist $S: W \to V$ with $ST = I_V$. $\textbf{Implications and Applications:}$ This characterization of injective linear maps is very useful in analyzing and understanding the behavior of linear transformations, especially in finite-dimensional settings. The existence of a right inverse map is a powerful tool for studying the properties of $T$, as it allows us to "undo" the action of $T$ on its range. This result has applications in areas like coding theory, where injective encoding maps are crucial for reliable data transmission and error correction. It also plays a role in the study of matrix invertibility and the development of algorithms for computing matrix inverses, which are essential in numerous scientific and engineering applications. From a theoretical perspective, the characterization connects to the broader study of invertible linear maps, which form an algebraic structure known as a general linear group. This group plays a fundamental role in abstract algebra, representation theory, and the study of symmetries in mathematics and physics.