Linear Algebra
Linear Algebra Done Right
Chapter 1 Vector Spaces
Exercises 1C
- For each of the following subsets of $\bf{F}^3$, determine whether it is a subspace of $\bf{F}^3$,
\[\begin{aligned} (a)& {(𝑥_1, 𝑥_2, 𝑥_3) \in \bf{F}^3 ∶ 𝑥_1 + 2𝑥_2 + 3𝑥_3 = 0} \\ (b)& {(𝑥_1, 𝑥_2, 𝑥_3) \in \bf{F}^3 ∶ 𝑥_1 + 2𝑥_2 + 3𝑥_3 = 4} \\ (c)& {(𝑥_1, 𝑥_2, 𝑥_3) \in \bf{F}^3 ∶ 𝑥_1𝑥_2𝑥_3 = 0} \\ (d)& {(𝑥_1, 𝑥_2, 𝑥_3) \in \bf{F}^3 ∶ 𝑥_1 = 5𝑥_3} \\ \end{aligned}\]