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Exercise 11.8.3

Avg difficulty 3.0

Which of the following formulas define an inner product on R3\mathbb{R}^{3}? Here, u=[u1,u2,u3]T\mathbf{u}=\left[u_{1},u_{2},u_{3}\right]^{T} and v=[v1,v2,v3]T\mathbf{v}=\left[v_{1},v_{2},v_{3}\right]^{T}.

  1. ⟨u,v⟩=u1v1+2u1v2+2u2v1+3u2v2+3u3v3\langle\mathbf{u},\mathbf{v}\rangle=u_{1}v_{1}+2u_{1}v_{2}+2u_{2}v_{1}+3u_{2}v_{2}+3u_{3}v_{3}.

  2. ⟨u,v⟩=u1v1−u1v2−u2v1+2u2v2+u3v3\langle\mathbf{u},\mathbf{v}\rangle=u_{1}v_{1}-u_{1}v_{2}-u_{2}v_{1}+2u_{2}v_{2}+u_{3}v_{3}.

  3. ⟨u,v⟩=u1v1−u1v2−u2v1+u2v2+4u3v3\langle\mathbf{u},\mathbf{v}\rangle=u_{1}v_{1}-u_{1}v_{2}-u_{2}v_{1}+u_{2}v_{2}+4u_{3}v_{3}.

  4. ⟨u,v⟩=u1v1−u1v2−u2v1+3u2v2−u2v3−u3v2+u3v3\langle\mathbf{u},\mathbf{v}\rangle=u_{1}v_{1}-u_{1}v_{2}-u_{2}v_{1}+3u_{2}v_{2}-u_{2}v_{3}-u_{3}v_{2}+u_{3}v_{3}.

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Positive semidefinite and positive definite matrices, Exercise 11.8.3

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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.