Vector Spaces Slides
Review of Finite Dimensional Vector Spaces

Definition:
A
vector space
is a set
with
and
.

Definition:
Linear Combination
or

Definition:
spans
, if

Definition:
are
linearly independent
if

Definition:
is
a
basis
for
if (1) they span
and (2)
are linearly independent.

Theorem:
Let
be a
basis for
, then

Definition:
are the coordinates of
with respect to the given basis for
.

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

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

Definition:
Inner Product
Measures how alike or parallel and are.

Definition:
Angle
between two vectors

Definition:
and
are
orthogonal
if
.

Definition:
Norm
of a vector
Measures the length of .

Definition:
Unit Vector

Definition:
Distance
between two points

Definition:
are
orthonormal
if
Geometric Representation of Signals
Let be an dimensional vector space of finite energy signals of duration .

Fact:
Let
be a basis
for
, then

Definition:
Inner product
and
norm
of signals

Fact:
are orthonormal
means

Definition:
Unit energy signal

Representation:
is
an orthonormal basis for
.
 Fact: Let and .