Graphics Reference
In-Depth Information
CHAPTER 1
Linear Algebra Topics
1.1
Introduction
This chapter assumes a basic knowledge and familiarity of linear algebra that is
roughly equivalent to what one would get from an introductory course in the subject.
See Appendix B and C for the necessary background material. In particular, we assume
the reader is familiar with the vector space structure of n-dimensional Euclidean
space
R
n
and its dot product and associated distance function. The object of this
chapter is to discuss some important topics that may not have been emphasized or
even covered in an introductory linear algebra course. Those readers with a weak
background in abstract linear algebra and who have dealt with vectors mostly in the
context of analytic geometry or calculus topics in
R
2
or
R
3
will also get a flavor of the
beauty of a coordinate-free approach to vectors. Proofs should not be skipped because
they provide more practice and insight into the geometry of vectors. The fact is that
a good understanding of (abstract) linear algebra and the ability to apply it is essen-
tial for solving many problems in computer graphics (and mathematics).
As in other places in this topic we have tried to avoid generality for generality's
sake. By and large, the reader can interpret everything in the context of subspaces of
R
n
; however, there are parts in this chapter where it was worthwhile to phrase the
discussion more generally. We sometimes talk about inner product spaces, rather than
just sticking to
R
n
and its dot product, and talk about vector spaces over other fields,
the complex numbers
C
in particular. This was done in order to emphasize the general
nature of the aspect at hand, so that irrelevant details do not hide what is important.
Vector spaces over the complex numbers will be important in later chapters.
Geometry is concerned with lots of different types of spaces. This chapter is about
the simplest of these, namely, the linear ones, and some related topics. Hopefully,
much of the material that is covered is review except that we shall approach the
subject here, like in many other places, with a vector approach. Sections 1.2-1.5
review the definition and basic properties of k-dimensional planes in
R
n
. We also look
at the abstract definition of angle and some important concepts related to ortho-
gonality, such as that of the orthogonal projection of a vector. Next, in Sections 1.6
and 1.7 we discuss the extremely important concepts of orientation and convexity.
