Let's start with a problem: are these polynomials linearly independent?

This introduces us to coordinate systems.

# Coordinate Systems

**coordinates**for x are the weights $c_1 ... c_n$ such that x = $c_1b_1 + ... + c_nb_n$.

## Theorem

**unique**set of scalars $c_1 ... c_n$ such that x = $c_1b_1 + ... + c_nb_n$. So, each point has a unique coordinate, just like in a normal 3d or 2d plane.

## Similar Theorem

A coordinate mapping is a one-to-one from V onto R^n.

# General Idea

When working in a vector space that is unorthodox, like polynomials, we can just morph it into real numbers, then it becomes easy.