Inner Product

If x and y are real numbers then the inner product is as follows:

$<x,y> := xy$

The result is simple multiplication. Actually we would also get this result with x and y both being vectors in 1D space.

Next, assume that instead of Scalars, we have Vectors with n components.

$\displaystyle \left\langle \begin{bmatrix} x_1 \\x_2 \\... \\x_n \end{bmatrix}, \begin{bmatrix} y_1 \\y_2 \\... \\y_n \end{bmatrix} \right\rangle := x^Ty = \sum_{i=1}^n x_i y_i = x_1y_1 + x_2y_2 + ... + x_ny_n$

Several Identities are true for Inner Products where u,v,w are vectors and a is a scalar:

<u + v,w> = <u,w> + <v,w>
<av,w> = a<v,w>

We recognize the above two as showining linearity for the first argument.

<v,w> = <w,v>
<v,v> > 0 if v is nonzero
<v,v> = 0 if v=0

We can see in these rules the same ideas that we saw for Bra-Ket notation.

Dot Product
Scalar Product
Cross Product

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