You now have every piece needed to read modern AI. Let's connect the dots explicitly — because everything you just learned is load-bearing in real research.

A neural network layer = output = W·x (matrix-vector multiply) followed by a simple nonlinearity. A deep network = many of those chained (matrix-matrix multiplies). Training just adjusts the numbers inside those weight matrices.

Where each idea shows up:
• Matrix multiply → every layer, every attention head in a transformer.
• Rank → model compression, LoRA fine-tuning, spotting redundant features.
• Eigenvectors/values → PCA and embeddings (finding the meaningful axes in data), and checking whether a recurrent network is stable.
• Inverse & null space → what information a model can and cannot recover.

🧠 And the same math describes the brain. Connectivity between brain regions is a matrix; population activity is a vector; how that activity evolves is governed by the matrix's eigenvalues. Linear algebra is the shared language of artificial and biological neural networks — which is exactly why it's the foundation of the whole NMA course.