On-Line Computer Graphics Notes

Vector Space Axioms

Overview

A Vector Space consists of a nonempty set of objects along with two algebraic operations -- addition and scalar multiplication. The space together with the operations must satisfy some straightforward axioms so that the ``mathematics'' that can be done with these objects make some sense. For example, we would expect that we can write expressions with these objects,

and that result not depend on which two objects we added first -- i.e. it should equal either or .

Formally, we can write down several properties that can be officially stated as the vector space axioms and they are all of the above form -- they are there so that the mathematics make sense. The axioms quite frequently have names associated with them (commutative, associative, distributive) as these properties are frequently associated with many abstract objects.

For a postscript version of these notes look here.

A Vector Space

So formally, A nonempty set of elements is called a vector space if in there are two algebraic operations (called addition and scalar multiplication), so that the following properties hold.

Addition Properties

Addition associates with every pair of elements and a unique element which is called the sum of and and is written . Addition satisfies the following :

• Commutativity -- for any two elements and in ,

  

  That is, it doesn't matter in which order two objects are added. The result is the same.

• Associativity -- for any three elements , and in ,

  

  So when summing three elements, it doesn't matter which two are added together first -- The result is the same.

• Zero Vector -- there is a unique element in called the zero element and denoted such that for every element

  

  That is, a vector space should have an additive identity, an element that when added to any other element does does nothing.

• Additive Inverse -- for each element , there is a unique element in , usually denoted -v, so that

  

  So, if we require a vector space to have a zero, then we should require it to have enough elements so that any element can be associated with another so that the two of them sum to the zero.

• The use of an additive inverse allows us to define a subtraction operation on elements. Simply

Scalar Multiplication Properties

Scalar Multiplication associates with every element and every scalar c, another unique element (usually written cv), For scalar multiplication the following properties hold:

• Distributivity -- for every scalar c and elements and in ,

  

  i.e. multiplication should work the way we are used to it working. If we multiply a quantity by a scalar, it is the same as multiplying all objects inside the quantity by the same scalar.

• Distributivity of Scalars -- for every two scalars and and element ,

  

  The same thing should work in reverse for members of the vector space. Multiplication by v should distribute to all scalars inside the parenthesis.

• Associativity -- for every two scalars and and element ,

  

  This says that, when we are dealing with a product of two scalars, it doesn't matter in what order we do the multiplication.

• Identity -- for every element ,

  

  The number 1 is the multiplicative identity for the set of scalars. It should then also have this property when being applied to members of the vector space.

We have taken some care here not to call the members of the vector space vectors, but elements. Vector is the common name when dealing with the vector space of three-dimensional vectors, but only confuses the issue when dealing with the vector space of quadratic polynomials.

Summary

The axioms of a vector space are constructed so that the operations will make sense to us considering the way that we normally do mathematics. However, they are general enough so that a large number of classes of objects can be considered vector spaces.

This document maintained by Ken Joy Comments to the Author All contents copyright (c) 1996, 1997 Computer Science Department, University of California, Davis All rights reserved.