Basis: Choosing Coordinates for a World

A vector is an object. Its coordinates are a description of that object relative to a chosen basis.

That distinction is easy to miss because in school we almost always use the standard basis $e_1=(1,0)$ and $e_2=(0,1)$ .

The same vector described using a Cartesian basis and a skew basis. — One geometric vector, two coordinate languages. Changing basis changes the numbers, not the vector.

What makes a basis?

A basis $\{b_1,\ldots,b_n\}$ must do two things:

span the space — every vector can be built from it;
be linearly independent — none of the basis vectors is redundant.

Then every vector has a unique representation

v=c_1b_1+\cdots+c_nb_n.

The tuple $(c_1,\ldots,c_n)$ is not the vector itself. It is the vector’s address in that coordinate system.

✦ A Connection Worth Noticing
Choosing a basis is like choosing vocabulary. The underlying thought may remain the same while its representation changes.

Change of basis

If $B$ is the matrix whose columns are the new basis vectors, then

[v]_{\text{standard}}=B[v]_B, \qquad [v]_B=B^{-1}[v]_{\text{standard}}.

A square coordinate grid beside a skew coordinate grid. — A basis does more than label axes: it determines the grid through which we describe the space.

Why this matters later

Eigenvectors give us a particularly revealing basis. Fourier analysis chooses sinusoidal basis functions. PCA chooses directions of maximal variance. Neural representations can be viewed through many coordinate systems.

The recurring lesson is:

coordinates are negotiable; structure is not.

Where next?

Continue with Linear Transformations to see what matrices do to an entire basis and grid.