Linear Algebra and the C Language/a0am

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>