.
On-Line Geometric Modeling Notes
These notes give the definition of a vector space and several of the
concepts related to these spaces. Examples are drawn from the vector
space of vectors in .
For a postscript version of these notes look here.
A nonempty set 
of elements 
is called a vector
space if in 
there are two algebraic operations (called
addition and scalar multiplication), so that the following
properties hold.
Addition associates with every pair of vectors 
and 
a
unique vector 
which is called the sum of 
and
and is written 
. In the case of the space of
2-dimensional vectors, the
summation is componentwise (i.e. if 
and 
, then 
), which can be
best illustrated by the ``parallelogram illustration'' below:
Addition satisfies the following :
and 
in 
,
, 
and 
in 
,
is calculated differently,
the result is the same.
called the zero
vector and denoted 
such that for every vector 
, there is a unique element
in 
, usually denoted 
, so that
is simply represented as
the vector of equal magnitude to 
, but in the
opposite direction.
The use of an additive inverse allows us to define a subtraction
operation on vectors. Simply 
The result of vector subtraction in the space of 2-dimensional
vectors is shown below.
Frequently this 2-d vectors is protrayed as joining the ends of the two original vectors. As we can see, since the vectors are determined by direction and length, and not position, the two vectors are equivalent.
Scalar Multiplication associates with every vector 
and every scalar c, another unique vector (usually written 
),
For scalar multiplication the following properties hold:
and 
in
,
and 
is just twice the vector
and 
and
vector 
,
and 
and
vector 
,
,
Examples of vector space abound in mathematics. The most obvious
examples are the usual vectors in 
, from which we have drawn
our illustrations in the sections above. But we frequently utilize
several other vectors spaces: The
3-d space of vectors, the
vector space of all polynomials of a fixed degree,
and vector spaces of
matrices. We briefly discuss these below.
The Vector Space of 3-Dimensional Vectors
The vectors in 
also form a vector space, where
in this case the vector operations of addition and scalar
multiplication are done componentwise. That is
and 
are vectors, then
addition is
and, if c is a scalar, scalar multiplication is given by
The axioms are easily verified (for example the additive identity of
is just
, and the zero vector is just 
.
Here the axioms just state what we always have been taught
about these sets of vectors.
The set of quadratic polynomials of the form
also form a vector space.
We add two of polynomials by adding their respective coefficients.
That is, if
and
, then
and
multiplication is done by multiplying the scalar by each
coefficient. That is, if s is a scalar, then
The axioms are again easily verified by performing the operations individually on like terms.
A simple extension of the above is to consider the set of polynomials of degree less than or equal to n. It is easily seen that these also form a vector space.
The set of 
Matrices form a vector space. Two
matrices can be added componentwise, and a matrix can be multiplied by
a scalar. All axioms are easily verified.
Given a vector space 
, the concept of a basis for the vector
space is fundamental for much of the work that we will do in computer
graphics. This section discusses several topics relating to linear
combinations of vectors, linear independence and bases.
Let 
be any vectors in a vector space 
and let 
be any set of scalars.
Then an expression of the form
is called a linear combination of the vectors.
This element is clearly a member of the vector space 
(just repeatedly
apply the summation and scalar multiplication axioms).
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.
It is straightforward to show that the span of any set of vectors is again a vector space.
Given a set of vectors 
from a vector space
. This set is called linearly independent in 
if the
equation
implies that 
for all i = 1, 2, ..., n.
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
This implies that 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
Any set of vectors containing the zero vector (
) is linearly
dependent.
To give an example of a linear independent set that everyone has seen,
consider the three vectors
in the vector space of vectors in 
Consider the equation
If we simplify
left-hand side by performing the operations componentwise and write
the right-hand side componentwise, we have
which can only be solved if 
.
Let 
be a set of vectors in a vector space
and let S be the span of 
. If 
is 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
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 i=1,2,...,n.
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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