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Back to Orthogonality

Exercise 11.2.1

Avg difficulty 3.0

Let A=[110120002]A=\left[\begin{array}{ccc}1&1&0\\ 1&2&0\\ 0&0&2\end{array}\right], and consider R3\mathbb{R}^{3} with the inner product given by ⟨u,v⟩=uTAv\langle\mathbf{u},\mathbf{v}\rangle=\mathbf{u}^{T}A\mathbf{v}. Which of the following vectors are orthogonal to each other?

u1=[111],u2=[−12−2],u3=[7−5−2],u4=[10−2−7].\mathbf{u}_{1}=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\quad\mathbf{u}_{2}=\left[\begin{array}{c}-1\\ 2\\ -2\end{array}\right],\quad\mathbf{u}_{3}=\left[\begin{array}{c}7\\ -5\\ -2\end{array}\right],\quad\mathbf{u}_{4}=\left[\begin{array}{c}10\\ -2\\ -7\end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Orthogonality, Exercise 11.2.1

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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.