FixTheRobot Explore Robotics / Linear Algebra for Robotics

Learning track

Linear Algebra for Robotics

The mathematics behind robot motion, perception and estimation, taught through robots instead of abstract exercises. From vectors to Lie groups, with interactive figures in every lesson and a graded lab in eight of the ten modules.

74 of 74 theory lessons and 8 of 8 labs are open. Items marked coming soon are not yet available. Start with M1-T1: Vectors and Coordinates. Read the resource survey · Jump to the syllabus

Before the course

Resources considered: the survey behind this course

Before writing a single lesson we surveyed 42 existing courses, syllabi, video lectures, textbooks, papers and tools. This is what we found, what we checked ourselves, and how it shaped the course. Every item below is a link; open any of them to learn from the original.

Show the survey: 42 resources, findings and licence labels

27 verified on the official page, 11 found by search and not read in full, 4 whose official page did not load when checked. We say which is which on every line, and we do not copy any of them: every lesson and lab here is original.

What the survey found

  • No single verified resource teaches the whole path from vectors to Lie groups with robotics from the first lesson. Michigan ROB 101 is closest for the beginning, Modern Robotics for the middle, and the Sola and Barfoot material for the end.
  • Several well-known "free" resources are free to read only. Of everything checked, only the Corke toolboxes (MIT) and Interactive Linear Algebra (GFDL, which forces the same licence) allow adaptation, so every lesson and lab here is written from scratch.
  • CMU 16-811 is mostly numerical methods, not a linear algebra course. Only about five of its 26 lectures are linear algebra.
  • Beginners are best served by computation and interaction rather than proof, so the course follows ROB 101 and Immersive Linear Algebra in spirit and lists Axler as optional reading.
  • Existing site content already teaches 2D and 3D transforms, quaternions, nonlinear least squares and the Kalman filter. This course links to those lessons instead of repeating them.

How to read the labels

Free to read
Free to access. No licence that allows copying or adapting. Linked, never mirrored.
Open, non-commercial
Creative Commons non-commercial licence (CC BY-NC or CC BY-NC-SA). May be adapted only for non-commercial use.
Open, adaptable
Licence that allows commercial adaptation, with conditions (MIT, or GFDL which requires the same licence).
No reuse terms stated
The page states a copyright but no reuse licence. Treated as read-only.
Not determined
The licence could not be confirmed.
Verified on the official page
We fetched the official page and the stated fact appeared on it.
Found by search, not read in full
A web search surfaced it; we have not read the whole page.
Official page did not load, not verified
The page returned an error (such as 503) when we checked, so details are unconfirmed.

Free to read does not mean free to copy or redistribute. Nothing here is mirrored on this site.

University courses and syllabi (16)

Where each syllabus was read for topic order and depth.

Video lectures (8)

Lecture videos for self-study alongside the course. Linked only; nothing is embedded or re-hosted.

  • ROB 101 lectures, Fall 2020 (YouTube playlist) University of Michigan Verified on the official pageOpen, non-commercial Licence: Course README: video is CC BY-NC 4.0 Full beginner linear algebra lecture series. Link taken from the course page; video contents not opened.
  • ROB 101 lectures, Fall 2021 (YouTube playlist) University of Michigan Verified on the official pageOpen, non-commercial Licence: Course README: video is CC BY-NC 4.0 Later offering of the same course. Link taken from the course repository; video contents not opened.
  • Modern Robotics specialization (six Coursera courses with lecture videos) Kevin Lynch, Northwestern University Verified on the official pageFree to read Licence: Coursera terms. Free audit is disputed between the Coursera FAQ and the Northwestern page Rigid-body motions, forward and inverse kinematics, velocity kinematics and statics.
  • Modern Robotics Course 2: Robot Kinematics Northwestern University Verified on the official pageFree to read Licence: Coursera terms Covers book chapters 4 to 7 per the Northwestern page.
  • MIT 18.06SC lecture videos (on the OCW course page) Gilbert Strang, MIT Verified on the official pageOpen, non-commercial Licence: CC BY-NC-SA 4.0 The standard university linear algebra lecture series.
  • Essence of Linear Algebra (video series) Grant Sanderson, 3Blue1Brown Verified on the official pageFree to read Licence: No open licence; clips under 60 seconds with attribution only Geometric intuition for matrices as transformations. The lesson titles could not be read from the page.
  • Introduction to Robotics (CS223A) lecture videos Oussama Khatib, Stanford University Found by search, not read in fullNot determined Licence: Not checked Kinematics lectures; a search listing names matrix algebra as the prerequisite.
  • Robot Academy (short lessons and masterclasses) Peter Corke, QUT Verified on the official pageFree to read Licence: Copyright page: "all rights reserved" unless marked otherwise; educational institutions may use it for teaching; individuals may save parts for their own study; no systematic downloading The site is online and filters lessons by Kinematics, Spatial Maths, Maths and Dynamics and Control. Individual lesson titles load dynamically and could not be read.

Textbooks and open books (13)

Free to read does not mean free to copy: the licence class says what is allowed.

  • Modern Robotics: Mechanics, Planning, and Control (free preprint PDF) Kevin Lynch and Frank Park Verified on the official pageFree to read Licence: No licence or reuse statement on the book page (preprint dated December 2019) Chapters 3 to 6: rigid-body motions, forward kinematics, velocity kinematics and statics (the Jacobian), inverse kinematics.
  • Modern Robotics book page (table of contents, slides, software) Northwestern University Verified on the official pageFree to read Licence: No licence statement Chapter list and companion material.
  • Modern Robotics software NxRLab Verified on the official pageNot determined Licence: Page states no licence Reference implementations for cross-checking.
  • Interactive Linear Algebra Dan Margalit and Joseph Rabinoff Verified on the official pageOpen, adaptable Licence: GNU Free Documentation License (adaptations must keep the same licence) A model for interactive figures. Not adapted: GFDL would force the same licence on our lessons.
  • Interactive Linear Algebra (Georgia Tech copy) Margalit and Rabinoff Official page did not load, not verifiedOpen, adaptable Licence: GFDL per the Duke page The Georgia Tech page returned HTTP 503; the Duke page was read instead.
  • Immersive Linear Algebra Jacob Strom, Kalle Astrom and Tomas Akenine-Moller Verified on the official pageNo reuse terms stated Licence: Page shows only a copyright line, no licence Ten chapters with interactive figures; a model for in-page interactivity.
  • Linear Algebra Done Right, 4th edition Sheldon Axler Verified on the official pageOpen, non-commercial Licence: Creative Commons BY-NC Proof-based optional reading.
  • Mathematics for Machine Learning Deisenroth, Faisal and Ong Verified on the official pageFree to read Licence: Copyright, PDF kept freely available; no reuse licence Matrix decompositions and PCA reading.
  • Introduction to Applied Linear Algebra (VMLS) Stephen Boyd and Lieven Vandenberghe Official page did not load, not verifiedNot determined Licence: Not checked Applied vectors, matrices and least squares. The official page returned HTTP 503.
  • VMLS Julia companion Boyd and Vandenberghe Found by search, not read in fullNot determined Licence: Not checked Companion code.
  • Introduction to Linear Algebra (publisher site) Gilbert Strang, Wellesley-Cambridge Press Found by search, not read in fullFree to read Licence: Commercial textbook The textbook behind MIT 18.06.
  • State Estimation for Robotics, 2nd edition (Cambridge), chapter 8 Matrix Lie Groups Timothy Barfoot Verified on the official pageFree to read Licence: Commercial book; the chapter page is paywalled (book purchase) Second edition, 2024: Part II Three-Dimensional Machinery, chapter 8 Matrix Lie Groups, pages 256 to 344. Reference for Modules 9 and 10.
  • State Estimation for Robotics, first edition (free author-hosted PDF) Timothy Barfoot Verified on the official pageFree to read Licence: Title page: "Copyright 2022", "Cambridge University Press is the Official Publisher", "This Unofficial Version Compiled on December 10, 2022". No reuse licence Free to read first edition with a revision history of corrections. Linked from the UCSD ECE276A page. Not the second edition.

Papers and tutorials (3)

Primary references for rotations, Lie groups and inverse kinematics.

Software used to cross-check this course (2)

Used in tests only. The labs themselves run a small pure-Python library inside your browser.

  • Robotics Toolbox for Python Peter Corke and Jesse Haviland Verified on the official pageOpen, adaptable Licence: MIT Independent check of kinematics results.
  • Spatial Maths for Python Peter Corke Verified on the official pageOpen, adaptable Licence: MIT Independent check of rotations and transforms.

The full analysis, including what each source changed in the course design, is in docs/LINEAR_ALGEBRA_RESOURCES.md in the repository. Links were last checked on 2026-11-10.

The course

Syllabus: 10 modules, 74 lessons, 8 labs

How the course works

Every module pairs lessons with practice. A lesson gives the geometric picture first, then the mathematics, a worked numeric example, and the robot application. Labs follow Problem, Experiment, Diagnose, Fix, Evaluate, Understand, and your code is graded on hidden randomised scenarios. You write Python in the page with a small linear algebra library, lakit, that is tested against NumPy.

Rotations, quaternions, nonlinear least squares and the Kalman filter are already taught in the Localization & SLAM and UAV tracks. This course links to those lessons and teaches the linear algebra underneath them.

Module 1

Mathematical Foundations

Vectors as the language of robot position, velocity and heading: how to add them, measure them, compare directions, and describe lines and planes.

Labs (1)

  1. M1-L12D Robot Vector NavigatorBrowser lab

Module 2

Matrices and Linear Transformations

A matrix is a table of numbers and a transformation of the plane. How matrices combine, what they do to shapes, what the determinant measures, and when a transformation can be undone.

Labs (1)

  1. M2-L1Matrix Transformation PlaygroundBrowser lab

Module 3

Linear Systems and Vector Spaces

Solving A x = b the way software does, and what rank and the four fundamental subspaces say about whether a robot problem has one answer, none, or infinitely many.

Labs (1)

  1. M3-L1Solve a Robot's ConstraintsBrowser lab

Module 4

Orthogonality and Projections

Perpendicularity as a tool: orthogonal bases, projecting onto lines and planes, the Gram-Schmidt process, orthogonal matrices and quadratic forms.

Labs (0)

Projection and orthogonality are practised in the least-squares lab (Module 5) and the PCA lab (Module 6).

Module 5

Least Squares and Numerical Methods

Fitting a model to noisy data, and why how you compute the fit matters as much as the formula: normal equations, QR, weights, regularisation and conditioning.

Labs (1)

  1. M5-L1LiDAR Plane FittingBrowser lab

Module 6

Eigenvalues, Eigenvectors and SVD

The directions a matrix leaves alone, the shape of its action, and the decomposition that reveals rank, conditioning and the best low-dimensional summary of data.

Labs (1)

  1. M6-L1Point Cloud PCABrowser lab

Module 7

Rotations and Rigid-Body Transformations

The linear algebra view of rotations and frames: rotation matrices, SO(3), homogeneous transforms, SE(3), composition and inverses. Quaternions and pose graphs live in the SLAM and UAV tracks.

Labs (1)

  1. M7-L1Robot Coordinate Frame ExplorerBrowser lab

Module 8

Jacobians and Robot Kinematics

From joint angles to hand motion: forward kinematics, the Jacobian as a local linear map, singularities, pseudoinverse and damped inverse kinematics, and null-space control of redundant arms.

Module 9

Covariance and State Estimation

Uncertainty as a matrix: covariance ellipses, Gaussians, how a linear map reshapes uncertainty, Mahalanobis distance, whitening and the Kalman filter as linear algebra.

Labs (1)

  1. M9-L1Robot Localization with UncertaintyBrowser lab

Module 10

Lie Groups and Advanced Robotics Mathematics

Why rotations are not a vector space, and the tools that replace addition: Lie groups and algebras, the exponential and logarithm maps, twists, perturbations and Jacobians on manifolds.

Labs (0)

Lie group ideas are explored with interactive figures inside the lessons. Pose-graph optimisation on these groups is practised in the SLAM track.