Numerical Methods in Computational Finance. Daniel J. Duffy
V left-parenthesis upper K right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column upper A 2 colon x plus y equals y plus x 3rd Row 1st Column Blank 2nd Column upper A 3 colon Exists unique 0 in upper V such that 0 plus x equals x plus 0 equals x 4th Row 1st Column Blank 2nd Column upper A 4 colon For each x in upper V there exists a unique y such that x plus y equals 0 5th Row 1st Column Blank 2nd Column left-parenthesis italic the negative of x right-parenthesis comma called negative x period EndLayout"/>
Axiom A1 states that addition is associative, and axiom A2 states that addition is commutative. The element 0 is called the zero element of the vector space.
Scalar multiplication is defined by the axioms
(4.3)
From these axioms we see that subtraction of vectors is possible because
The prototypical examples of vector spaces are n-dimensional vectors and rectangular matrices over a field K:
(4.4)
For matrices:
(4.5)
We now define an important non-negative real-valued function on a vector space V called a norm. It has the following properties:
(4.6)
Some examples of norms for two-dimensional vectors are:
The following norms for vectors and matrices are used in applications:
(4.7)
(4.8)
Whereas the norm is a measure of the size of a vector, it is also possible to find the distance between two vectors. A natural way to proceed is, given a set X to define a real-valued function on