On-Line Computer Graphics Notes
Overview
Given any set of elements of a
space 
, we can form linear combinations of these elements and
obtain their span, which is again a
vector space contained in 
. If it happens that the span of the
elements is
, then we can write any element of 
as a linear combination
of some elements from this set. If the elements of the set themselves are
linear independent, then we say
that they form a basis. Discovering a basis
of a vector space is most important, as we then can work mostly with
the basis, and extend our results trivially to the entire vector space
by using linear combinations.
For a postscript copy of these notes look here.
What is a Basis?
Let 
be a set of vectors in a vector space
and let S be the span of 
. If 
are linearly independent, then we say that these vectors form a basis
for S and S has dimension n. Since these vectors span S, any
vector 
can be written uniquely as a linear combination of
these elements
The uniqueness follows from the argument that if there were two such representations
then by subtracting the two equations, we obtain
which can only happen if all the expressions 
are zero,
since the vectors 
are assumed to be
linearly independent. Thus we necessarily have that 
for
all 
.
If S is the entire vector space 
, we say that
forms a basis for 
, and 
has
dimension n.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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