# Vector Spaces

## Review of Finite Dimensional Vector Spaces

**Definition:**An -dimensional

*vector space*is a set of objects along with a definition of how to add two elements of and a definition of how to scale an element by a real number. The example to keep in mind is the one you are already familiar with-three dimensional Euclidean space ( ).

**
Definition:
**
*
Linear Combination
*

or |

**
Definition:
**
A set
of
vectors in
is said to
*
span
*
, if every vector
in
can be written as a linear combination of the elements in the set.

**
Definition:
**
A set
of
vectors in
is called
*
linearly independent
*
if the equation

holds only for . Otherwise they are

*linearly dependent*.

**
Definition:
**
A set
of
vectors in
is called a
*
basis
*
for
if they span
and
are linearly independent.

**
Theorem:
**
If the set
is a
basis for
then in the representation of

the coefficients are unique. The numbers are called the coordinates of with respect to the given basis for .

**
Definition:
**
The
*
dimension
*
of a vector space
is the
number of elements in a basis for
.

**
Definition:
**
Let us construct the unique
*
coordinate vector
*
corresponding to the element
in
.

Note that the coordinate vector ``lives'' in -dimensional Euclidean space regardless of the nature of the vector space . For example, might be a vector space of polynomials. In digital communications, is a space of functions.