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." The symbolism for an inner product consists of two vectors separated by a common and imposed by two parentheses. The inner product is as follows: (u,v).
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
The inner product also lies in a vector space that can be represented by V. This is called an inner product space. An inner product space is defined simply as a [1] vector space that contains a inner product. As a side note, if the vector space is to the nth power it is refered to as an [2] Euclidean space which is a finite space as well.
The inner product is useful in computing various other items in mathematics as well. By knowing the inner product, one can then in turn figure out the angle between two vectors. This simple equation for determining the angle, theta, is given by:
cos(θ)=
(u,v)
----
||u|| ||v||
where the denominator consists of the product of the lengths of vectors u and v.
- It may also be helpful to look at other explanations of inner products. These links will bring you to other people explaining inner products:
[3] The standard inner product
[4] More on inner products
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
