On-Line Computer Graphics Notes

Examples of Vector Spaces

The simplest example of a vector space to understand is the space of vectors in the 2-dimensional plane. Most students clearly have more experience with this space than with any other example, and pictures can be easily generated which illustrate the axioms.

In reality, this space, when equipped with a dot product operation, is an inner-product space, and in this case we can talk about ``orthogonal'' vectors and the length of vectors -- concepts for which we are also familiar.

Herein we present the axioms for this vector space in some detail, with illustrations provided for each of the concepts. Our vectors in this case are denoted in their classic form, as letters with an arrow on the top (e.g. ).

The set of vectors in 2-dimensional space forms a vector space, and so has two algebraic operations: addition and scalar multiplication. Addition associates with every pair of vectors and a unique vector which is called the sum of and and is written . In this case the summation is componentwise (i.e. if and , then ), which can be best illustrated by the ``parallelogram illustration'' below:

Addition satisfies the following :

• Commutativity -- for any two vectors and in ,

  

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

  

  This rule is illustrated in the figure below. One can see that even though the sum is calculated differently, the result is the same.

  

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

  

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

  

  is simply represented as the vector of equal magnitude to , but in the opposite direction.

  

  The use of an additive inverse allows us to define a subtraction operation on vectors. Simply The result of vector subtraction in the space of 2-dimensional vectors is shown below.

  

  Frequently this 2-d vector is protrayed as joining the ends of the two original vectors. As we can see, since the vectors are determined by direction and length, and not position, the two vectors are equivalent.

  

Scalar Multiplication associates with every vector and every scalar c, another unique vector (usually written ),

For scalar multiplication the following properties hold:

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

  

  In the case of 2-dimensional vectors, this can be easily seen by extending the parallelogram illustration above. We can see below that the sum of the vectors and is just twice the vector

  

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

  

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

  

• Identity -- for every vector ,

  

In summary, the set of vectors in two-dimensional space form a vector space. The axioms of a vector space are, in most cases, represented geometrically.

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.