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Linear Algebra and the C Language/a0am
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Dot Product
In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number.... Wikipedia: Dot product
Some Properties
Properties by axioms:
<u,v> = <v,u> ............................ Symmetry axiom
<u+v,w> = <u,w>+<v,w> ............ Additivity axiom
<ku,v> = k<u,v> ......................... Homogeneity axiom
<u,u> = 0 if and only if u = 0 ...... Positivity axiom
Properties by defintion:
<0,v> = <v,0> = 0
<u,v+w> = <u,v>+<u,w>
<u,v-w> = <u,v>-<u,w>
<u-v,w> = <u,w>-<v,w>
k<u,v> = <u,kv>
Properties:
Cauchy-Schwarz : |<u,v>| <= ||u|| ||v||
2||u||^2 + 2||v||^2 = ||u + v||^2 + ||u - v||^2
||u||^2 + ||v||^2 - ||u - v||^2 = 2 * <u,v>