# Highly Nonlinear Approximations for Sparse Signal Representation

## Frames and bases for finite a dimension vector space

If
are some elements of a vector space ,
by a **linear combination** of
we mean an
element in of the form
,
with
.

Let be a subset of element of . The set of all *linear
combinations* of elements of is called the **span** of
and is denoted by
.

A subset
of is said to be **linearly independent**
if and only if

**linearly dependent**if it is not linearly independent.

is said to be a **basis** of if it is
linearly independent and
. The **dimension**
of a finite dimensional vector space is the number of elements in
a basis for . The number of elements in a set is termed the
**cardinality** of the set.

If the number of elements spanning a finite dimensional vector space
is larger than the number of elements of a basis for the
same space, the set is called a **redundant frame**. In other words a
redundant frame (henceforth called simply a frame)
for a finite dimensional vector space is a
linearly dependent set of vectors spanning .

Let be a spanning set for . Then every can be expressed as