WHY: If you record 50 neurons, you might expect 50-dimensional data. But neurons are correlated — groups fire together. The data actually only lives in a few independent dimensions. Knowing how many truly independent directions exist is the foundation of PCA.

Think of a 50-piece orchestra. Each musician could theoretically play anything independently. But they follow the same conductor and sheet music — so instead of 50 independent sounds, there are maybe only 4 sections (strings, brass, woodwind, percussion) moving somewhat independently. 50 musicians, but only ~4 independent dimensions of variation.

Same with neurons. Groups fire together because they receive similar inputs and are connected. The data might have 50 neurons but only move along 3, 5, or 8 truly independent directions.

Span: all the points you can reach using linear combinations of a set of vectors.

Linearly independent: two vectors are independent if neither is just a scaled copy of the other. They point in genuinely different directions.

Linearly dependent: one IS a scaled copy — they carry redundant information.

Example: two neurons that ALWAYS fire at exactly double each other's rate.

Neuron A = [1, 2, 3], Neuron B = [2, 4, 6] = 2 × Neuron A

They are linearly dependent — they always move together. Together they only span a 1D line, not 2D space. You're not getting independent information from both.

Real neural data almost always lives on a low-dimensional subspace — far fewer dimensions than neurons. This is why PCA works. The brain is not using all available dimensions.