Linear Algebra Done Right · Problem 1C.1

Finite-Dimensional Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · subspaces

Problem

For each of the following subsets of

\mathbf{F}^3

, determine whether it is a subspace of

\mathbf{F}^3

. ((a))

\{(x_1, x_2, x_3) \in \mathbf{F}^3 : x_1 + 2x_2 + 3x_3 = 0\}

((b))

\{(x_1, x_2, x_3) \in \mathbf{F}^3 : x_1 + 2x_2 + 3x_3 = 4\}

((c))

\{(x_1, x_2, x_3) \in \mathbf{F}^3 : x_1 x_2 x_3 = 0\}

((d))

\{(x_1, x_2, x_3) \in \mathbf{F}^3 : x_1 = 5x_3\}