Module 2 Group Activity
1
Matrix Theory
1. For each of the following, determine whether \( U \) is a subspace of \( V \). You may assume the typical addition and scalar multiplication for \( V \). Make sure to justify your answers!
(a) \( V=\mathbb{C}^{3} \) as a complex vector space, \( U=\mathbb{C}^{2} \)
(b) \( V=\mathbb{R}^{2 \times 2} \) as a real vector space, \( U=\left\{A \in \mathbb{R}^{2 \times 2}: A^{2}=0\right\} \)
(c) \( V=\mathcal{P}_{2}(\mathbb{C}) \) as a complex vector space, \( U=\left\{a+b t+c t^{2}: a, b, c \in \mathbb{C}, a=-2 c\right\} \)
(d) \( V=\mathcal{C}([0,1], \mathbb{R}) \) as a real vector space, \( U=\{f \in V: f(1 / 2) \geq 0\} \).
Note. \( U \) is the set of functions \( f \in V \) such that \( f \) evaluated at \( 1 / 2 \) is positive. For example, consider \( f_{1}(t)=t+1 \) and \( f_{2}(t)=t^{2}-1 \). \( f_{1}, f_{2} \in V \) as both are continuous, real-valued functions on \( [0,1] \). However, \( f_{1} \in U \) since \( f_{1}(1 / 2)=3 / 2 \geq 0 \) while \( f_{2} \notin U \) since \( f_{2}(1 / 2)=-3 / 4<0 \).
