## Changing Basis

The use of coordinates to define vectors implies an agreement about which Basis Vectors we use. In 2d space, we commonly use the standard basis vectors:

$\hat{i}=\begin{bmatrix}1 \\ 0\end{bmatrix}$, $\hat{j}=\begin{bmatrix}0 \\ 1\end{bmatrix}$.

However, we are technically not limited to just the standard basis vectors: we can use any set of vectors to describe a coordinate system. Perhaps we encounter an Alien whose coordinate system uses these basis vectors:

$\hat{e}_{1} = \begin{bmatrix}2 \\ 4\end{bmatrix}$, $\hat{e}_{2} = \begin{bmatrix}1 \\ 1\end{bmatrix}$.

If the Alien describes a vector, say $\begin{bmatrix}3 \\ 1\end{bmatrix}$, in their coordinate system, we'd have first to translate it to our system.

We can translate back into our system by creating a matrix which uses the Alien basis vectors as the columns:

$\begin{bmatrix}2 && 1 \\ 4 && 1\end{bmatrix}\begin{bmatrix}3 \\ 1\end{bmatrix} = \begin{bmatrix}7 \\ 13\end{bmatrix}$

We can think of that as a Matrix Transformation that scales basis vector $\hat{e}_1$ by $3$ and $\hat{e}_2$ by $1$.

We can convert a vector described in our coordinate system to the Alien using the Matrix Inverse of our Alien's basis vector matrix:

$\begin{bmatrix}2 && 1 \\ 4 && 1\end{bmatrix}^{-1}\begin{bmatrix}7 \\ 13\end{bmatrix} = \begin{bmatrix}3 \\ 1\end{bmatrix}$

If we wish to perform a transformation described in our Basis, for example, a rotational transformation on an alternate basis, we can follow these steps:

- Convert the vector into our Basis by applying our friend's transformation matrix.
- Perform the translation.
- Convert the vector back into our friend's Basis by applying the inverse of the transformation.

In notation, if we have vector $\vec{v}$ described in our friend's Basis $A$, we can apply Matrix Transformation $M$, described in our Basis, as follows:

$A^{-1}MA \ \vec{v}$