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.