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# Dot product

nosplit "Linear algebra" words: 177
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 case.
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 : $$M=⎣⎢⎡​100​010​001​⎦⎥⎤​ (32)$$ so that: $$x⋅y​=[x1​​x2​​x3​​]⎣⎢⎡​100​010​001​⎦⎥⎤​⎣⎢⎡​y1​y2​y3​​⎦⎥⎤​=x1​y1​+x2​y2​+x3​y3​ (33)$$