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 pdf 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 :

**Commutativity**- for any two vectors and in ,**Associativity**- for any three vectors , and in ,**Zero Vector**- there is a unique vector in called the*zero vector*and denoted such that for every vector**Additive Inverse**- for each element , there is a unique element in , usually denoted , so thatThe 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 , another unique vector (usually written
),

For scalar multiplication the following properties hold:

**Distributivity**- for every scalar and vectors and in ,**Distributivity of Scalars**- for every two scalars and and vector ,**Associativity**- for every two scalars and and vector ,**Identity**- for every 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

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

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 . 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.

This element is clearly a member of the vector space (just repeatedly apply the summation and scalar multiplication axioms).

The set
that contains all possible linear combinations of
is called the * span* of
. We frequently say that
is *
spanned* (or * generated*) by those
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

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

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

Consider the equation

A Basis for a Vector Space

Let be a set of vectors in a vector space and let be the span of . If is linearly independent, then we say that these vectors form a basis for and has dimension . Since these vectors span , any vector can be written uniquely as

then by subtracting the two equations, we obtain

If is the entire vector space , we say that forms a basis for , and has dimension .

2000-11-28