On-Line Computer Graphics Notes

Examples of Vector Spaces

Examples of vector space abound in mathematics. The most obvious examples are the usual vectors in , from which we have drawn our illustrations in the sections above. But we frequently utilize several other vectors spaces: The 3-d space of vectors, the vector space of all polynomials of a fixed degree, and vector spaces of matrices. We briefly discuss these below.

  The Vector Space of 3-Dimensional Vectors

The vectors in also form a vector space, where in this case the vector operations of addition and scalar multiplication are done componentwise. That is and are vectors, then addition is

and, if c is a scalar, scalar multiplication is given by

The axioms are easily verified (for example the additive identity of is just , and the zero vector is just . Here the axioms just state what we always have been taught about these sets of vectors.

  Vector Spaces of Polynomials

The set of quadratic polynomials of the form

also form a vector space. We add two of polynomials by adding their respective coefficients. That is, if and , then

and multiplication is done by multiplying the scalar by each coefficient. That is, if s is a scalar, then

The axioms are again easily verified by performing the operations individually on like terms.

A simple extension of the above is to consider the set of polynomials of degree less than or equal to n. It is easily seen that these also form a vector space.

  Vector Spaces of Matrices

The set of Matrices form a vector space. Two matrices can be added componentwise, and a matrix can be multiplied by a scalar. All axioms are easily verified.

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.