Graphics Reference
In-Depth Information
APPENDIX C
Basic Linear Algebra
C.1
More on Linear Independence
Vector spaces were already defined in Appendix B. This appendix gives a highly con-
densed summary of all the important facts about vector spaces that are used in the
topic. For more details the interested reader is referred to any topic on linear algebra,
such as those listed in the bibliography. For simplicity, unless stated otherwise, all
vector spaces here are assumed to be finite-dimensional vector spaces over the
reals.
Related to the notion of linearly independent vectors is the notion of linearly
independent points.
Definition. Elements p 0 , p 1 ,..., p k of a vector space are said to be linearly inde-
pendent points if
ppp p
-
,
-
, ... ,
p p
-
1
0
2
0
k
0
are linearly independent as vectors; otherwise, they are linearly dependent points .
C.1.1. Theorem. Whether or not points are linearly independent or dependent does
not depend on the order in which they are listed.
Proof.
This is an easy exercise.
In Figure C.1(a), the points p 0 , p 1 , and p 2 are linearly independent points, but not
in Figure C.1(b). Intuitively speaking, points are linearly independent if they generate
a maximal dimensional space (maximal with respect to the number of points
involved). In Figure C.1(a) and (b) the points generate two- and one-dimensional sub-
spaces, respectively. It is because three points can generate a two-dimensional space
that the points in Figure C.1(b) are called linearly dependent.
Sometimes one wants to decompose a vector space into a sum of subspaces.
Search WWH ::




Custom Search