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Primarily, it is necessary to begin with the basic definitions of Inner Products and Orthogonality. An inner product is defined by Bernard Kolman in his Elementary Linear Algebra book as being "a function V that assigns to each ordered pair of vectors u,v in V a real number (u,v) satisfying the following properties." There are four properties taken from Elementary Linear Algebra book that inner products must follow: | Primarily, it is necessary to begin with the basic definitions of Inner Products and Orthogonality. An inner product is defined by Bernard Kolman in his Elementary Linear Algebra book as being "a function V that assigns to each ordered pair of vectors u,v in V a real number (u,v) satisfying the following properties." There are four properties taken from Elementary Linear Algebra book that inner products must follow: | ||
Revision as of 09:20, 7 December 2011
== Inner Products and Orthogonality
==
Primarily, it is necessary to begin with the basic definitions of Inner Products and Orthogonality. An inner product is defined by Bernard Kolman in his Elementary Linear Algebra book as being "a function V that assigns to each ordered pair of vectors u,v in V a real number (u,v) satisfying the following properties." There are four properties taken from Elementary Linear Algebra book that inner products must follow:
1) (u,u) is greater than or equal to 0 ((u,u)=0 if u equals the zero vector)
2) (v,u)=(u,v) for an u,v in V
3) (u+v,w)=(u,w)+(v,w) for an u,v,w in V
4) (cu,v)=c(u,v) for u, v in V and c a real scalar
Simply, as follows in the book is a definition for Orthogonality. "Two vectors u and v in V are orthogonal if (u,v)=0." This is to say that given one vector crossed with another vector is equal to zero, then they are orthogonal. For example using variables: u=[a;b] v=[c;d]
(u,v)=(u x v) = ac + bd = 0 => orthogonal vectors
For example using numbers:
u=[1;0] v=[0;1]
(u,v)=(u x v) = 1(0) + 0(1) = 0 => orthogonal vectors
