the determinant tells you about volume scaling under a linear transformation

Eigenvalues and eigenvectors

Eigenvectors are unchanged in direction by a linear transformation

Rotation matrix

Determinant of a 2x2 matrix: ad - bc

Principal component analysis

Eigenvectors point along maximum variance

PCA vs t-SNE: PCA preserves global variance linearly, t-SNE preserves local structure nonlinearly

How can we teach computers to understand what we like?

the trace equals the sum of eigenvalues: tr(A) = Σλ_i

How can a vector stay the same after a transformation?

Neural scaling law

Chinchilla scaling law: optimal model size scales linearly with compute budget