linear-algebra

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.