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Exercise 11.1.1

Avg difficulty 3.0

For each matrix AA, determine whether the formula ⟨u,v⟩=uTAv\langle\mathbf{u},\mathbf{v}\rangle=\mathbf{u}^{T}A\mathbf{v} determines an inner product on R2\mathbb{R}^{2}.

(a) A=[1002],(b) A=[3113],(c) A=[1102],(d) A=[100−1],(e) A=[1111].(a)~{}A=\left[\begin{array}{cc}1&0\\ 0&2\end{array}\right],\quad(b)~{}A=\left[\begin{array}{cc}3&1\\ 1&3\end{array}\right],\quad(c)~{}A=\left[\begin{array}{cc}1&1\\ 0&2\end{array}\right],\quad(d)~{}A=\left[\begin{array}{cc}1&0\\ 0&-1\end{array}\right],\quad(e)~{}A=\left[\begin{array}{cc}1&1\\ 1&1\end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Real inner product spaces, Exercise 11.1.1

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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.