# Highly Nonlinear Approximations for Sparse Signal Representation

## Vector Space

A **vector space** over a field
is
a set
together with two operations vector addition, denoted
for
and
scalar multiplication, denoted
for
and
,
such that the following axioms are satisfied:

- .
- , .
- There exists an element , called the zero vector, such that , .
- There exists an element , called the additive inverse of , such that , .
- and .
- and .
- and .
- , , where 1 denotes the multiplicative identity in .

**vectors**.