--- $\textbf{Exercise 18 Commentary:}$ This exercise establishes necessary and sufficient conditions for the existence of an injective (one-to-one) linear map between two finite-dimensional vector spaces $V$ and $W$. Specifically, it shows that there exists an injective linear map $T: V \to W$ if and only if $\text{dim}(V) \leq \text{dim}(W)$. The proof utilizes the Fundamental Theorem of Linear Maps, similar to Exercise 17. If $T: V \to W$ is injective, then $\text{dim}(\text{null}(T)) = 0$, which implies $\text{dim}(V) = \text{dim}(\text{range}(T)) \leq \text{dim}(W)$, giving the necessary condition. Conversely, if $\text{dim}(V) \leq \text{dim}(W)$, the proof constructs an explicit injective linear map $T$ by mapping basis vectors of $V$ injectively into $W$, using the Linear Map Lemma (Theorem 3.4). This shows that the dimension inequality is also sufficient for the existence of an injective linear map. $\textbf{Exercise 18 Examples:}$ 1) There exists an injective linear map from $\mathbb{R}^2$ to $\mathbb{R}^3$, since $\text{dim}(\mathbb{R}^2) = 2 \leq 3 = \text{dim}(\mathbb{R}^3)$. One such map is $T(x, y) = (x, y, 0)$. 2) There does not exist an injective linear map from $\mathbb{C}^4$ to $\mathbb{C}^3$, since $\text{dim}(\mathbb{C}^4) = 4 > 3 = \text{dim}(\mathbb{C}^3)$. 3) There exists an injective linear map from $\mathcal{P}_2(\mathbb{R})$ to $\mathcal{P}_3(\mathbb{R})$, since $\text{dim}(\mathcal{P}_2(\mathbb{R})) = 3 \leq 4 = \text{dim}(\mathcal{P}_3(\mathbb{R}))$. One such map is the inclusion map $T(a + bx + cx^2) = a + bx + cx^2$. 4) There exists an injective linear map from $M_{2,2}(\mathbb{R})$ to $\mathbb{R}^4$, since $\text{dim}(M_{2,2}(\mathbb{R})) = 4 \leq 4 = \text{dim}(\mathbb{R}^4)$. One such map sends a $2 \times 2$ matrix to the vector obtained by stacking its columns. $\textbf{Implications and Applications:}$ This result is the dual of Exercise 17 and is equally fundamental in linear algebra. It characterizes when it is possible to "embed" a lower-dimensional vector space into a higher-dimensional one via an injective linear map, preserving the linear structure and distinctness of vectors. Injective linear maps play a crucial role in the study of subspaces, linear independence, and the analysis of linear transformations modulo their ranges. They are essential in areas like signal processing, where injective maps model sampling and reconstruction operations that aim to faithfully represent lower-dimensional signals in higher-dimensional spaces without losing information. The result also connects to the theory of linear codes and the construction of efficient error-correcting codes, where injective maps are used to embed message spaces into higher-dimensional code spaces. From a geometric perspective, injective linear maps allow us to understand embeddings of lower-dimensional objects into higher-dimensional spaces, which is important in computer graphics, computer vision, and many areas of pure mathematics. --- $\textbf{Exercise 18 Implications and Applications (continued):}$ The characterization of the existence of injective linear maps between finite-dimensional vector spaces in terms of their dimensions has important implications and applications in various areas of mathematics, physics, and engineering. In signal processing, injective linear maps model sampling and reconstruction operations that aim to faithfully represent lower-dimensional signals in higher-dimensional spaces without losing information. The existence condition provided in this exercise is crucial for designing sampling schemes that can capture the full information content of the original signal. In the study of linear codes and the construction of efficient error-correcting codes, injective maps are used to embed message spaces into higher-dimensional code spaces. The existence condition in this exercise is essential for understanding the coding capabilities and error-correction properties of these codes, as well as for analyzing the existence of decoding algorithms. In the analysis of subspaces, linear independence, and the study of linear transformations modulo their ranges, injective linear maps play a fundamental role. The existence condition connects to the properties of these subspaces and the ability to represent vectors in the domain space uniquely within the range space. In computer graphics and computer vision, injective linear maps are used to understand embeddings of lower-dimensional objects into higher-dimensional spaces, such as embedding 2D curves or surfaces into 3D space. The result in this exercise provides insights into the conditions under which such embeddings can be achieved without losing the distinctness of the original objects. Furthermore, this result has implications in the study of linear systems and the analysis of linear transformations in control theory and dynamical systems. Injective maps are used to model the propagation of information or signals through linear systems, and the existence condition is crucial for understanding the observability and controllability properties of these systems. Overall, the characterization of the existence of injective linear maps between finite-dimensional vector spaces is a fundamental result in linear algebra that has wide-ranging implications and applications in various fields, including signal processing, coding theory, subspace analysis, computer graphics, and the study of linear systems and dynamical processes.