# 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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