--- $\textbf{Exercise 33 Commentary:}$ 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 $T_\mathbb{C}(u + iv) = Tu + iTv$. (a) The first part verifies that $T_\mathbb{C}$ is indeed a complex linear map by checking the additivity and homogeneity conditions. (b) The second part 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 $Tv = 0$, which implies $u = v = 0$ if $T$ is injective. Conversely, if $Tu = 0$, then $T_\mathbb{C}(u) = 0$, so $T_\mathbb{C}$ is injective if $T$ is. (c) The third part 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 $\text{range}(T) = \mathbb{R}$. 3) 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 $\text{range}(T) = \mathbb{C}^2$. $\textbf{Implications and Applications:}$ The complexification construction has important implications in the study of linear maps and their representations, as well as in the development of efficient algorithms for computing null spaces, ranges, and other properties of matrices. This construction allows for the application of tools and techniques from complex linear algebra to real linear maps, which can often lead to simpler computations and more elegant theoretical results. It also plays a crucial role in the study of complex vector spaces, which are fundamental in areas such as quantum mechanics, signal processing, and control theory. The complexification of a linear map 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. From a theoretical perspective, the complexification construction sheds light on the interplay between real and complex linear algebra, and the connections between these two domains. It also has applications in the study of algebraic and geometric structures, such as Lie groups and Lie algebras, which play a fundamental role in various areas of mathematics and physics.