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Exercise 11.10.3

Avg difficulty 3.0

Suppose u,v∈Cn\mathbf{u},\mathbf{v}\in\mathbb{C}^{n} and u⋅u=1\mathbf{u}\boldsymbol{\cdot}\mathbf{u}=1, v⋅v=2\mathbf{v}\boldsymbol{\cdot}\mathbf{v}=2, and u⋅v=2i\mathbf{u}\boldsymbol{\cdot}\mathbf{v}=2i. Then compute:

  1. (2u+v)⋅(u−v)(2\mathbf{u}+\mathbf{v})\boldsymbol{\cdot}(\mathbf{u}-\mathbf{v}).

  2. (iu−v)⋅(u−2iv)(i\mathbf{u}-\mathbf{v})\boldsymbol{\cdot}(\mathbf{u}-2i\mathbf{v}).

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Complex inner product spaces, Exercise 11.10.3

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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.