Change of Basis / Coordinate Systems

A basis is a set of vectors that can be used as a coordinate system for a vector space. The standard basis we're all used to is ($\mathbf{\hat{\imath}}$, $\mathbf{\hat{\jmath}}$), which are the vectors $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ respectively. The coordinates of a vector $\mathbf{v} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ simply mean "3 steps in the $\mathbf{\hat{\imath}}$ direction and 2 steps in the $\mathbf{\hat{\jmath}}$ direction."

However, we can choose any two linearly independent vectors to be our basis. A **change of basis** is the process of re-expressing a vector in terms of a new set of basis vectors. It's like translating a description of a location from one language (e.g., "3 steps east, 2 steps north") to another language (e.g., "1.5 steps northeast, 0.5 steps northwest"). The location itself doesn't change, just how we describe it.

The Change of Basis Matrix

If we have a new basis B = {b₁, b₂}, we can create a "change of basis matrix" $P_B$ whose columns are these new basis vectors. To find the standard coordinates of a vector given in the new basis, we simply multiply:

To go the other way—from standard coordinates to the new basis—we use the inverse matrix:

This is extremely powerful in quantitative finance, particularly in Principal Component Analysis (PCA), where we change the basis of our data to a new set of orthogonal axes (the eigenvectors of the covariance matrix) that represent the directions of maximum variance.