--- COMMENTARY: $\textbf{Exercise 5.}$ Give an example of $T \in \mathcal{L}(\mathbb{R}^4)$ such that $\operatorname{range} T = \operatorname{null} T$. $\textbf{Solution 5.}$ Define $T \in \mathcal{L}(\mathbb{R}^4, \mathbb{R}^4)$ by $T(x_1, x_2, x_3, x_4) = (x_3, x_4, 0, 0).$ Then $\operatorname{range} T = \operatorname{null} T = {(x_3, x_4, 0, 0) \in \mathbb{R}^4 : x_3, x_4 \in \mathbb{R}}$. $\textit{Commentary:}$ This exercise asks for an example of a linear operator on $\mathbb{R}^4$ whose range and null space are equal. The example given in the solution is a projection onto the first two coordinates, followed by a shift of coordinates. Geometrically, $T$ maps $\mathbb{R}^4$ onto the $x_1x_2$-plane (which is the range of $T$), and also maps any vector with zero first two coordinates to zero (which is the null space of $T$). Thus, the range and null space are both the $x_1x_2$-plane. This is an example of a nilpotent operator, meaning that some power of $T$ is zero. In this case, $T^2 = 0$, because $T$ maps everything into its null space. Such operators are important in various contexts, such as in the: Jordan canonical form theorem, which describes the structure of nilpotent operators. $\textit{Examples:}$ 1. Define $T \in \mathcal{L}(\mathbb{R}^5)$ by $T(x_1, x_2, x_3, x_4, x_5) = (x_2, x_3, x_4, 0, 0)$. Then $\operatorname{range} T = \operatorname{null} T = {(x_2, x_3, x_4, 0, 0) : x_2, x_3, x_4 \in \mathbb{R}}$. 2. Define $T \in \mathcal{L}(\mathbb{C}^6)$ by $T(z_1, z_2, z_3, z_4, z_5, z_6) = (z_4, z_5, z_6, 0, 0, 0)$. Then $\operatorname{range} T = \operatorname{null} T = {(z_4, z_5, z_6, 0, 0, 0) : z_4, z_5, z_6 \in \mathbb{C}}$. 3. Define $T \in \mathcal{L}(\mathbb{F}_2^4)$ by $T(x_1, x_2, x_3, x_4) = (x_3, 0, 0, 0)$. Then $\operatorname{range} T = \operatorname{null} T = {(x_3, 0, 0, 0) : x_3 \in \mathbb{F}_2}$. These examples demonstrate that the phenomenon of a linear operator having equal range and null space can occur over different fields and in different dimensions. In each case, the operator is a kind of shifted projection, mapping the space onto a subspace and also mapping the complement of that subspace to zero.