Mr. Integral

Understand systems and structure

Linear Algebra

Connect equations with geometry, from vectors and matrices to eigenvalues, least squares and SVD.

Start learning 6 units · 19 lessons

The course, unit by unit.

Open a unit for its introduction, map and lesson route.

01Vectors, Matrices and SystemsBuild and solve systems, then factor matrices for repeated solves.3 lessons

What does a system of equations look like geometrically?

Start with the introduction & map →
Unit cheat sheet All the key ideas in one reference
  1. 01Vectors, Linear Combinations and Systems
  2. 02Elimination, Pivots and Free Variables
  3. 03Matrix Products, Inverses and LU
02Vector SpacesExplain span, independence, bases and the four fundamental spaces.3 lessons

How many independent directions does a model really have?

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Unit cheat sheet All the key ideas in one reference
  1. 01Span, Independence and Bases
  2. 02Subspaces, Null Spaces and Column Spaces
  3. 03Rank, Nullity and the Four Fundamental Spaces
03Linear TransformationsRead geometric maps, change coordinates and measure volume scaling.3 lessons

Is the vector changing, or only the coordinates used to describe it?

Start with the introduction & map →
Unit cheat sheet All the key ideas in one reference
  1. 01Linear Maps and Geometry
  2. 02Change of Basis and Similarity
  3. 03Determinants, Orientation and Invertibility
04Orthogonality and Least SquaresProject, construct orthonormal bases and fit inconsistent systems.3 lessons

What is the closest answer when an exact fit is impossible?

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Unit cheat sheet All the key ideas in one reference
  1. 01Dot Products and Orthogonal Projection
  2. 02Gram–Schmidt and QR Factorization
  3. 03Least Squares and Data Fitting
05Eigenvalues and DynamicsFind invariant directions and connect them to powers and quadratic forms.4 lessons

Which directions keep their identity as a system evolves?

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Unit cheat sheet All the key ideas in one reference
  1. 01Eigenvalues and Eigenspaces
  2. 02Diagonalization and Discrete Dynamics
  3. 03Complex Numbers and Oscillatory Modes
  4. 04Symmetric Matrices and Quadratic Forms
06SVD and Engineering ComputationUnderstand low rank, pseudoinverses and sensitivity; check your understanding.3 lessons

Which parts of a matrix carry the strongest signal?

Start with the introduction & map →
Unit cheat sheet All the key ideas in one reference
  1. 01Singular Value Decomposition
  2. 02Pseudoinverses, Low Rank and Conditioning
  3. 03Linear Algebra Checkpoint