Inner products

Algebra Level 4

Which of the following are inner products?

I. On R2, {\mathbb R}^2, with vectors x,yx,y written as 2×1 2\times 1 column vectors, define x,y=xTAy, \langle x,y \rangle = x^T A y, where A=(0121). A = \begin{pmatrix} 0&-1 \\ 2&1 \end{pmatrix}.

II. Same as in I, but A=(1221). A = \begin{pmatrix} 1&2 \\ 2&1 \end{pmatrix}.

III. The Minkowski product: on R4, {\mathbb R}^4, with vectors v=(x1,y1,z1,t1) {\bf v} = (x_1,y_1,z_1,t_1) and w=(x2,y2,z2,t2), {\bf w} = (x_2,y_2,z_2,t_2), write v,w=x1x2+y1y2+z1z2t1t2.\langle {\bf v},{\bf w} \rangle = x_1x_2+y_1y_2+z_1z_2-t_1t_2.

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