# 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:

1. .
2. , .
3. There exists an element , called the zero vector, such that , .
4. There exists an element , called the additive inverse of , such that , .
5. and .
6. and .
7. and .
8. , , where 1 denotes the multiplicative identity in .
The elements of a vector space are called vectors.