Inner products

Algebra Level 5

Which of the following are inner products?

I. On \( {\mathbb R}^2,\) with vectors \(x,y\) written as \( 2\times 1\) column vectors, define \( \langle x,y \rangle = x^T A y \) where \( A = \begin{pmatrix} 0&-1 \\ 2&1 \end{pmatrix}.\)

II. Same as in I, but \( A = \begin{pmatrix} 1&2 \\ 2&1 \end{pmatrix}.\)

III. The Minkowski product: on \( {\mathbb R}^4,\) with vectors \( {\bf v} = (x_1,y_1,z_1,t_1) \) and \( {\bf w} = (x_2,y_2,z_2,t_2),\) write \[ \langle {\bf v},{\bf w} \rangle = x_1x_2+y_1y_2+z_1z_2-t_1t_2. \]

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