![[sol-5.pdf#page=19]] --- $\textbf{Exercise 1.}$ Explain why each linear functional is surjective or is the zero map. $\textbf{Solution 1.}$ A linear functional is a linear map into $\mathbb{F}$. The range of a linear map is a subspace of the target space. The only subspaces of $\mathbb{F}$ are $\mathbb{F}$ and $\{0\}$. Thus each linear functional is surjective (if the range equals $\mathbb{F}$) or is the zero map (if the range equals $\{0\}$). --- $\textbf{Exercise 1.}$ Explain why each linear functional is surjective or is the zero map. $\textbf{Solution 1.}$ A linear functional is a linear map into $\mathbb{F}$. The range of a linear map is a subspace of the target space. The only subspaces of $\mathbb{F}$ are $\mathbb{F}$ and $\{0\}$. Thus each linear functional is surjective (if the range equals $\mathbb{F}$) or is the zero map (if the range equals $\{0\}$). $\textit{Commentary:}$ This exercise demonstrates the special nature of linear functionals, which are linear maps from a vector space to its underlying field. Because the field has only two subspaces, a linear functional can only be surjective or the zero map. This is in contrast to linear maps between general vector spaces, which can have ranges of various dimensions. $\textit{Example:}$ Consider the linear functional $\varphi: \mathbb{R}^2 \to \mathbb{R}$ defined by $\varphi(x, y) = x$. This functional is surjective because for every $a \in \mathbb{R}$, there exists a vector $(a, 0) \in \mathbb{R}^2$ such that $\varphi(a, 0) = a$. On the other hand, the linear functional $\psi: \mathbb{R}^2 \to \mathbb{R}$ defined by $\psi(x, y) = 0$ for all $(x, y) \in \mathbb{R}^2$ is the zero map.