Question

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 $.

          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 \( f_{2}(1 / 2)=-3 / 4<0 \).

$Module 2 Group Activity 1 Matrix Theory 1. For each of the following, determine whether U is a subspace of V. You may assume the typical addition and scalar multiplication for V. Make sure to justify your answers! (a) V=ℂ^3 as a complex vector space, U=ℂ^2 (b) V=ℝ^2 × 2 as a real vector space, U={A ∈ℝ^2 × 2: A^2=0} (c) V=𝒫2(ℂ) as a complex vector space, U={a+b t+c t^2: a, b, c ∈ℂ, a=-2 c} (d) V=𝒞([0,1], ℝ) as a real vector space, U={f ∈ V: f(1 / 2) ≥ 0}. Note. U is the set of functions f ∈ V such that f evaluated at 1 / 2 is positive. For example, consider f1(t)=t+1 and f2(t)=t^2-1. f1, f2∈ V as both are continuous, real-valued functions on [0,1]. However, f1∈ U since f1(1 / 2)=3 / 2 ≥ 0 while f2∉ U since f2(1 / 2)=-3 / 4<0.$

Added by Eduardo J.

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