Linear Algebra

This page is a reference manual, not a full course. Come back to the relevant section whenever a main-track module stalls you.

Minimal knowledge set

• Vector spaces and bases: the state vector $s_t$ and observation vector $o_t$ are points in vector spaces; a change of basis is a change of coordinate system.
• Matrix factorization: eigendecomposition explains the modes of linear dynamics (in $s_{t+1} = A s_t$, the eigenvalues of $A$ determine stability or divergence); SVD powers dimensionality reduction and least squares.
• Linear least squares: the update step of the Kalman filter is essentially recursive weighted least squares.
• Coordinate transforms and homogeneous coordinates: everything in Track B's camera model $K[R|t]$ and rigid-body transforms rests on these.

When to consult

Main-track module	Linear algebra used
Track A 03 State Space Models	Matrix multiplication, covariance matrices, eigenvalues and stability
Track A 04 Representation Learning	Rank, spectra, geometry of embedding spaces
Track B 02 Camera and Geometry	Homogeneous coordinates, projection matrix factorization
Track B 06 State Estimation	Jacobians, least squares, covariance propagation

Best external resources

• 3Blue1Brown, Essence of Linear Algebra: the first choice for geometric intuition — watch this first.
• Gilbert Strang, MIT 18.06 (MIT OCW): a systematic course with a textbook.
• Mathematics for Machine Learning (Deisenroth et al., mml-book.github.io), Chapters 2–4: a crash course in linear algebra from an ML perspective; free PDF.

Self-check

You are ready when you can compute the eigenvalues of a $2\times2$ matrix by hand, explain the geometric meaning of SVD, and write the vector expression for point-to-plane distance.

Next

Back to the main tracks: Track A Module 03 or Track B Module 02