Basis and Dimension

You've been working with vectors in terms of the standard axes: right and up. But those axes aren't special — they're just a convenient choice. Any two non-parallel vectors can serve as a basis.

What is a basis?

A basis for ℝ² is a set of two vectors where:

• They are linearly independent — neither is a scalar multiple of the other
• They span ℝ² — you can reach any point using linear combinations of them

The standard basis is $\mathbf{e}_1 = [1, 0]$ and $\mathbf{e}_2 = [0, 1]$. But $[2, 1]$ and $[0, 3]$ also form a valid basis — just a tilted, stretched one.

Coordinates are relative

When we say a vector is $[3, 2]$, we mean: 3 steps in the e₁ direction, 2 steps in the e₂ direction. Change the basis, and the same physical arrow gets different coordinates.

Dimension

The dimension of a space is the number of vectors in any basis for it. ℝ² has dimension 2. ℝ³ has dimension 3. A line through the origin has dimension 1 — one vector spans it.

A key fact: every basis for ℝ² has exactly 2 vectors. You can't span the plane with 1 vector, and 3 vectors are always redundant (one is a combination of the others).

Try it

Drag the blue and pink vectors to change the basis. The purple vector v is always expressed as a combination of the two basis vectors — watch the coefficients update.

Now try dragging e₁ and e₂ on top of each other. The basis becomes degenerate: they no longer span the plane.

import numpy as np
P = np.array([[2, 0], [1, 3]])  # columns = basis vectors
v = np.array([4, 3])
coords = np.linalg.solve(P, v)  # coordinates in new basis
print(coords)  # [2, 0.33...]