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Exercise 5.3.1

Avg difficulty 3.0

Which of the following sets are subspaces of R3\mathbb{R}^{3}? Explain.

  1. V1={u=[u1u2u3] ∣ ∣u1∣≤4}V_{1}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}\left|u_{1}\right|\leq 4\right\}.

  2. V2={u=[u1u2u3] ∣ ui≥0 for each i=1,2,3}V_{2}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}\text{$u_{i}\geq 0$ for each $i=1,2,3$}\right\}.

  3. V3={u=[u1u2u3] ∣ u3+u1=2u2}V_{3}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}u_{3}+u_{1}=2u_{2}\right\}.

  4. V4={u=[u1u2u3] ∣ u3≥u1}V_{4}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}u_{3}\geq u_{1}\right\}.

  5. V5={u=[u1u2u3] ∣ u3=u1=0}V_{5}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}u_{3}=u_{1}=0\right\}.

Selinger, Matrix Theory and Linear Algebra, Spans, linear independence, and bases in Rn, Subspaces of R^n, Exercise 5.3.1

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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.