Editing Inner product space (section)

===Euclidean vector space===

More generally, the [[Real coordinate space|real <math>n</math>-space]] <math>\R^n</math> with the [[dot product]] is an inner product space, an example of a [[Euclidean vector space]].
<math display=block>
\left\langle
  \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix},
  \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}
\right\rangle 
= x^\textsf{T} y = \sum_{i=1}^n x_i y_i = x_1 y_1 + \cdots + x_n y_n,
</math>
where <math>x^{\operatorname{T}}</math> is the [[transpose]] of <math>x.</math>

A function <math>\langle \,\cdot, \cdot\, \rangle : \R^n \times \R^n \to \R</math> is an inner product on <math>\R^n</math> if and only if there exists a [[Symmetric matrix|symmetric]] [[positive-definite matrix]] <math>\mathbf{M}</math> such that <math>\langle x, y \rangle = x^{\operatorname{T}} \mathbf{M} y</math> for all <math>x, y \in \R^n.</math> If <math>\mathbf{M}</math> is the [[identity matrix]] then <math>\langle x, y \rangle = x^{\operatorname{T}} \mathbf{M} y</math> is the dot product. For another example, if <math>n = 2</math> and <math>\mathbf{M} = \begin{bmatrix} a & b \\ b & d \end{bmatrix}</math> is positive-definite (which happens if and only if <math>\det \mathbf{M} = a d - b^2 > 0</math> and one/both diagonal elements are positive) then for any <math>x := \left[x_1, x_2\right]^{\operatorname{T}}, y := \left[y_1, y_2\right]^{\operatorname{T}} \in \R^2,</math>
<math display=block>\langle x, y \rangle 
:= x^{\operatorname{T}} \mathbf{M} y 
= \left[x_1, x_2\right] \begin{bmatrix} a & b \\ b & d \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} 
= a x_1 y_1 + b x_1 y_2 + b x_2 y_1 + d x_2 y_2.</math> 
As mentioned earlier, every inner product on <math>\R^2</math> is of this form (where <math>b \in \R, a > 0</math> and <math>d > 0</math> satisfy <math>a d > b^2</math>).