linear-algebra

Every renderer answers one question first: at this pixel, which surface is visible? Getting there means moving geometry through five coordinate spaces. This lesson builds that chain, explains why a fourth coordinate is not a trick, and shows where depth precision quietly goes wrong.

Building the machinery: the linear space of allowed directions at a point, the inner product that gives it geometry, and why the Riemannian gradient is the ambient gradient projected rather than a new derivative.

Rotations, subspaces, low-rank matrices and covariances are not vectors in flat space. Why treating those constraints as penalties or projections wastes structure, and what it means to say the search space is a manifold.

The mathematical object behind a qubit — a complex unit vector in a two-dimensional Hilbert space — and why measurement collapses superposition. The structural difference between a quantum state and a classical bit, expressed in math.

Vectors are more than arrows — they're the atoms of every ML model, physics engine, and signal processor alive. Build rock-solid intuition for linear combinations, span, independence, basis, and orthogonality, then verify it all in NumPy.

When there's no exact solution, project. When data is high-dimensional, compress. The SVD is the Swiss Army knife that does both — and more. Master orthogonal projection, the normal equations, the Singular Value Decomposition, low-rank approximation, and the pseudoinverse.

A matrix doesn't just hold numbers — it reshapes space. Master the geometric view of matrix-vector multiplication, the four fundamental subspaces, rank, the determinant as a volume-scaling factor, and invertibility — all grounded in NumPy.

Some vectors only get scaled by a matrix — they don't rotate at all. These eigenvectors reveal the skeleton of a linear transformation. Master the eigen-equation, the characteristic polynomial, diagonalization, and why eigenstructure powers PCA, PageRank, and stability analysis.

Move beyond single-input PID to the state-space framework: the state vector, matrix dynamics, controllability, pole placement via state feedback, and LQR — the tool that scales to full robot arms and drones.