--- COMMENTARY: $\textbf{Exercise 33 Commentary:}$ This exercise explores the properties of the "complexification" of a real linear map $T: V \to W$, which is a complex linear map $T_\mathbb{C}: V_\mathbb{C} \to W_\mathbb{C}$ defined by $T_\mathbb{C}(u + iv) = Tu + iTv$. Part (a) verifies that $T_\mathbb{C}$ is indeed a complex linear map, by checking additivity and homogeneity. Part (b) shows that $T_\mathbb{C}$ is injective if and only if $T$ is injective. The key idea is that if $T_\mathbb{C}(u+iv) = 0$, then $Tu = 0$ and $Tv = 0$, so $u = v = 0$ if $T$ is injective. Conversely, if $Tu = 0$, then $T_\mathbb{C}(u) = 0$, so $T_\mathbb{C}$ is injective if $T$ is. Part (c) shows that $\text{range}(T_\mathbb{C}) = W_\mathbb{C}$ if and only if $\text{range}(T) = W$. This follows from the fact that every complex vector in $W_\mathbb{C}$ is of the form $w_1 + iw_2$ for some $w_1, w_2 \in W$, and $T_\mathbb{C}$ hits all such vectors if and only if $T$ hits all of $W$. $\textbf{Exercise 33 Examples:}$ 1. Let $V = \mathbb{R}^2, W = \mathbb{R}^3$, and let $T: V \to W$ be the linear map given by $T(x,y) = (x+2y, 3x-y, 2x+4y)$. Then $T_\mathbb{C}: \mathbb{C}^2 \to \mathbb{C}^3$ is the linear map given by $T_\mathbb{C}(z_1, z_2) = (z_1 + 2z_2, 3z_1 - z_2, 2z_1 + 4z_2)$. One can check that $T_\mathbb{C}$ is injective and $\text{range}(T_\mathbb{C}) = \mathbb{C}^3$, since $T$ is injective and $\text{range}(T) = \mathbb{R}^3$. 2. Let $V = \mathcal{P}_2(\mathbb{R}), W = \mathbb{R}$, and let $T: V \to W$ be the linear map given by $T(a + bx + cx^2) = a$. Then $T_\mathbb{C}: \mathcal{P}_2(\mathbb{C}) \to \mathbb{C}$ is the linear map given by $T_\mathbb{C}(p(x)) = p(0)$. One can check that $T_\mathbb{C}$ is not injective, and $\text{range}(T_\mathbb{C}) = \mathbb{C}$, since $T$ is not injective and $\text{range}(T) = \mathbb{R}$. 1. Let $V = \mathbb{C}^3, W = \mathbb{C}^2$, and let $T: V \to W$ be the linear map given by the matrix $\begin{pmatrix} 1 & i & 0 \\ 0 & 1 & 1 \end{pmatrix}$. Then $T_\mathbb{C} = T$ is simply the same linear map, but viewed as a complex linear map. One can check that $T_\mathbb{C}$ is injective, and $\text{range}(T_\mathbb{C}) = \mathbb{C}^2$, since $T$ is injective and $\text{range}(T) = \mathbb{C}^2$. 2.