--- COMMENTARY: $\textbf{Exercise 20 Commentary:}$ This is the dual result to Exercise 19, characterizing surjective linear maps in terms of the existence of a left inverse. A linear map $T : V \to W$ is surjective if and only if there exists $S : W \to V$ such that $TS$ is the identity on $W$. This gives a critera for determining whether $T$ hits every element of the codomain $W$. $\textbf{Exercise 20 Examples:}$ 1. Let $T : \mathbb{R}^3 \to \mathbb{R}^2$ be the linear map 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 $TS = I_W$.