On-Line Computer Graphics Notes

## Vector Spaces

One of the most fundamental concepts in mathematics is that of a vector space (or equivalently a linear space). This ``label'', when given to a class of objects, indicates that the class has a structure. We know for example that two members of the class can be combined through an addition process, or they can be scaled by multiplication. We can form linear combinations of these objects, and can search for bases If it satisfies the properties of an inner-product space, then we can talk about ``orthogonal'' elements and orthogonal bases, and this gives a tremendous structure to the class.

This is one of the first abstractions that a student sees in university-level mathematics -- usually associated with a beginning linear algebra course -- and in our experience the typical student decides that this is just a generalization of vectors in 3-d space''. In some sense this is correct, and the 3-d vectors can be a canonical example of a vector space. However, associating this concept with only the 3-d vectors is looking only at the tip of a very large iceberg as this concept is also useful for spaces of matrices, spaces of functions, spaces of operators, and many more.

In these notes we review the properties of a vector space, including linear independence and bases. We also review several detailed examples of vector spaces. The reader should review the axioms and be aware of the manipulations with basis elements, but should also carefully review the examples, as these provide much of the foundation for mathematical work in computer graphics and geometric modeling.

Ken Joy Mon Dec 9 08:46:24 PST 1996