WHY: Normal brain data uses 'neuron space' — each axis is one neuron. PCA finds a BETTER basis where each axis represents a pattern of population activity. Same data, different coordinate system — suddenly the structure becomes visible.

The standard basis in 2D is the axes you already know:

• e₁ = [1, 0] — the x-axis
• e₂ = [0, 1] — the y-axis

When you say a point is at (4, 2), you mean: 4 units along e₁, 2 units along e₂. You've been using a basis all along.

A basis is any set of vectors that:
1. Are linearly independent (no redundancy)
2. Span the entire space (you can reach any point)

You need exactly N basis vectors for N-dimensional space — no more, no less.

Vector space: a set of vectors closed under linear combinations — adding or scaling any vectors in the set always gives you another vector still in the set. ℝ¹, ℝ², ℝ³ are all vector spaces.

Subspace: a smaller space inside a bigger one. Example: a plane through the origin inside 3D space is a valid subspace (linear combinations of vectors on the plane always stay on the plane). A plane NOT through the origin is NOT a subspace.

Neural meaning: population activity typically lives on a low-dimensional subspace of the full neuron space. PCA finds the axes of that subspace.

Switching to a better basis = PCA. Same data points, different axes — axes that align with how the population actually varies. A cleaner picture of what the brain is doing.