1. Dimensions of Subspaces. \( (6=2+2+2) \) For each part below, I give you a finite-dimensional vector space \( V \) and a subspace \( W \subset V \). Give me (i) the dimension of \( V \), (ii) the dimension of \( W \), (iii) a basis \( \beta_{W} \) of \( W \), (iv) a basis \( \beta_{V} \) of \( V \) such that \( \beta_{W} \subset \beta_{V} \).
(a) \( V=\mathcal{F}\left(\mathbf{F}_{3}, \mathbf{R}\right) \) and \( W=\left\{f \in \mathcal{F}\left(\mathbf{F}_{3}, \mathbf{R}\right): f(\overline{0})=f(\overline{1})=f(\overline{2})\right\} \), as a vector space over \( \mathbf{R} \).
(b) \( V=\mathbf{C}^{2} \) as a vector space over \( \mathbf{R} \), and \( W=\left\{\left(z_{1}, z_{2}\right): \bar{z}_{1}=-z_{2}\right\} \).
(c) Fix \( m \in \mathbf{R} . V=\mathbf{R}^{2} \), better known as the \( x y \)-plane, and \( W \) is the graph of \( f(x)=m x \).
