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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: | ||
| − | 1) (u,u) is greater than or equal to 0 ((u,u)=0 if u equals the zero vector) | + | '''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 | 2) (v,u)=(u,v) for an u,v in V | ||
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3) (u+v,w)=(u,w)+(v,w) for an u,v,w 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 | + | 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. | 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: | For example using variables: | ||
| − | + | ''' | |
u=[a;b] v=[c;d] | u=[a;b] v=[c;d] | ||
(u,v)=(u x v) = ac + bd = 0 => orthogonal vectors | (u,v)=(u x v) = ac + bd = 0 => orthogonal vectors | ||
| − | For example using numbers: | + | '''For example using numbers:''' |
u=[1;0] v=[0;1] | u=[1;0] v=[0;1] | ||
Revision as of 08:18, 7 December 2011
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
