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Linear Algebra: Structure, Space and Transformation

A rigorous first course in linear algebra for university study and serious self-directed learners: systems and elimination, vector spaces and linear maps, orthogonality and least squares, determinants, eigenvalues and the singular value decomposition — with computation, geometry and proof given equal weight.

≈ 4 h 25 min1 sections · 4 lessonsNo ratings yet
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What you'll learn

  • Solve and classify linear systems, express every solution parametrically, and interpret solution sets geometrically as intersections and affine subsets.
  • Use subspaces, span, independence, basis, dimension, rank and nullity precisely in computation, definition and proof.
  • Represent a linear transformation by a matrix, change basis, and reason about kernel, image, injectivity and surjectivity.
  • Apply inner products, orthogonal complements, Gram-Schmidt, QR and least squares to projection and approximation problems.
  • Compute and interpret determinants, eigenvalues, eigenvectors and diagonalisation, including the spectral theorem for symmetric matrices.
  • State the geometric role of the singular value decomposition and use low-rank approximation to summarise a matrix.
  • Write concise mathematical arguments, construct counterexamples, and identify the first invalid step in a flawed proof.

Before you start

  • Confident algebraic manipulation: fractions, exponents, rearranging equations, and solving simultaneous linear equations
  • Coordinate geometry in the plane, including the equation of a line
  • Familiarity with basic proof language: if/then, converse, counterexample

Academic references

Use the latest available edition and your institution's prescribed text.

  • OpenStax, Calculus, Volumes 1–3
  • Gilbert Strang, Introduction to Linear Algebra
  • Daniel J. Velleman, How to Prove It

Formal work standard

  • State definitions and hypotheses before using a theorem.
  • Separate proof, counterexample, and numerical evidence.
  • Complete the problem sheet without solution steps before reviewing errors.

Course content

0/4 lessons
1. Vectors, Linear Combinations and Geometry4 lessons
  1. Vectors in ℝⁿ: Components, Magnitude and AlgebraLesson55 min
  2. Linear Combinations and the Idea of SpanLesson60 min
  3. Lines and Planes in Vector and Parametric FormLesson60 min
  4. Section 1 Consolidation: Vectors and GeometryLesson90 min