--- $\textbf{Exercise 20 Commentary:}$ This exercise is the dual of Exercise 19, providing a characterization of surjective linear maps in terms of the existence of a "left inverse" map. Specifically, it shows that a linear map $T: V \to W$ is surjective if and only if there exists a linear map $S: W \to V$ such that $TS$ is the identity map on $W$. The proof follows a similar structure to Exercise 19. If $T$ is surjective, the proof constructs a left inverse $S$ by mapping basis vectors of $W$ to preimages in $V$ under $T$. Conversely, if $TS$ is the identity on $W$, then for any $w \in W$, we have $w = (TS)w \in \text{range}(T)$, showing that $T$ is surjective. $\textbf{Exercise 20 Examples:}$ 1) Let $T: \mathbb{R}^3 \to \mathbb{R}^2$ be given by $T(x, y, z) = (x+2y, 3x-z)$. Then $T$ is surjective, with left inverse $S(a, b) = (a/4, (b+3a)/4, (3a-b)/4)$. 2) Let $V = \mathcal{P}_3(\mathbb{R})$ and $W = \mathcal{P}_2(\mathbb{R})$, and let $T: V \to W$ be differentiation. Then $T$ is surjective, with left inverse $S(a+bx+cx^2) = \int_0^x (a+bt+ct^2)\,dt$. 3) Let $V = \mathbb{F}_3^2$ and $W = \mathbb{F}_3^3$, and let $T: V \to W$ be the linear map with matrix $\begin{pmatrix}1&2\\2&1\\1&0\end{pmatrix}$. Then $T$ is not surjective, and there does not exist $S: W \to V$ with $TS = I_W$. $\textbf{Implications and Applications:}$ The characterization of surjective linear maps in terms of left inverses is a powerful tool for analyzing the behavior of linear transformations and their ranges. It provides a way to construct explicit preimages of vectors in the codomain, which is useful in areas like inverse problems, control theory, and signal processing. This result also plays a crucial role in the study of matrix invertibility and the development of algorithms for computing matrix inverses. It connects to the broader study of invertible linear maps, which form an algebraic structure known as the general linear group, and has applications in areas such as coding theory, cryptography, and the study of symmetries in mathematics and physics. From a theoretical perspective, the characterization sheds light on the duality between injective and surjective linear maps, and the interplay between their null spaces and ranges. It also connects to the study of quotient spaces and the analysis of linear transformations modulo their ranges, which has applications in areas like dynamical systems and differential equations.