Coordinate Systems

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

$$1 + t^3, 3 + t - 2t^2, -t + 3t^2 - t^3$$ There's no obvious way of doing this, but we can convert to real number vectors. Let $< a,b,c,d>$ equal $a + bx + cx^2 + dx^3$. So, we can rewrite our vectors as $$\begin{bmatrix}1 & 3 & 0\\0 & 1 & -1\\0 & -2 & 3\\1 & 0 & -1\end{bmatrix} \rightarrow \begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}$$ It has a pivot in each column, so it's linearly independent.

This introduces us to coordinate systems.

Coordinate Systems

Suppose $\mathcal{B}$ is a basis for V and x is in V. The coordinates for x are the weights $c_1 ... c_n$ such that x = $c_1b_1 + ... + c_nb_n$.

Theorem

If $\mathcal{B}$ is a basis for a vector space V, then for all x in V, there exists a 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.