On-Line Computer Graphics Notes
The properties of a vector space allow us to add vectors and multiply vectors by scalars. By repeatedly doing this process we produce the linear combination
where 
are any elements in a vector space 
and 
are a set of scalars.
A linear combination of this form is clearly
a member of the vector space 
-- since each 
is a
member, and if we sum two members, we create another member.
These linear combinations are the most common operation on a vector
space. A frequent concern is to attempt to find the elements 
so
that any member of the vector space can be written as a linear
combination in the above form. Such a set of elements is called a
basis.
The Span of a set of Vectors
Given a set of elements 
from a vector space
, then
the set S that contains all possible linear combinations of
is called the span of
. We frequently say that S is
spanned (or generated) by those n vectors.
The span S of any set of elements from 
is again a
vector space, under the same operations of 
. This is easy to
see, because if you have two elements 
and 
from the span
S, then if 
and
, then
which is another linear combination of the 
elements and is in
S, and for a scalar d
which is also a linear combination of the 
elements and is in
S.
These two together imply that linear combinations of elements of S are again in S. The other axioms of a vector space are also satisfied by S and are easily checked.
Linear Independence
If we are given a set
of elements 
from a vector space
, this set is called linearly independent in 
if the
equation
implies that 
for all 
.
If a set of vectors is not linearly independent, then it is called
linearly dependent. This implies that the equation above has a
nonzero solution, that is there exist 
which are
not all zero, such that
In this case at least one of the vectors 
can be written in terms
of the other n-1 vectors in the set. Assuming that 
is not
zero, we can see that
and this characterizes linear dependence.
It's clear that
any set of vectors containing the zero vector (
) is linearly
dependent.
In summary, linear combinations are the complex calculation
that can be done in a vector space. Given any set of elements of a
space 
, we can form the span of these elements, which is again a
vector space contained in 
. If the span of the elements is
, and the elements themselves are linear independent, then we say
that they form a basis -- which implies that
any element in 
can be written as a linear combination of this
set.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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