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Exercise 5.3.2

Avg difficulty 3.0

Let w∈R4\mathbf{w}\in\mathbb{R}^{4} be a given fixed vector. Let

M={u=[u1u2u3u4]∈R4 ∣ w⋅u=0}.M=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\\ u_{4}\end{array}\right]\in\mathbb{R}^{4}~{}\right|~{}\mathbf{w}\boldsymbol{\cdot}\mathbf{u}=0\right\}.

Is MM a subspace of R4\mathbb{R}^{4}? Explain.

Selinger, Matrix Theory and Linear Algebra, Spans, linear independence, and bases in Rn, Subspaces of R^n, Exercise 5.3.2

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.