The definition of the "dot product" of a general space varies quite a lot with different contexts.

Most definitions tend to be bilinear forms.

We use the unqualified generally refers to the dot product of Real coordinate spaces, which is a positive definite symmetric bilinear form. Other important examples include:The rest of this section is about the $R_{n}$ case.

- the complex dot product, which is not strictly symmetric nor linear, but it is positive definite
- Minkowski inner product, sometimes called" "Minkowski dot product is not positive definite

The positive definite part of the definition likely comes in because we are so familiar with metric spaces, which requires a positive norm in the norm induced by an inner product.

The default Euclidean space definition, we use the matrix representation of a symmetric bilinear form as the identity matrix, e.g. in $R_{3}$:
so that:

$M=⎣⎢⎡ 100 010 001 ⎦⎥⎤ $

$x⋅y =[x_{1} x_{2} x_{3} ]⎣⎢⎡ 100 010 001 ⎦⎥⎤ ⎣⎢⎡ y_{1}y_{2}y_{3} ⎦⎥⎤ =x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}$