Linear Algebra for Research

This module develops the linear-algebraic vocabulary and computational fluency needed to formulate and solve the matrix problems that underpin regression, dimensionality reduction, graph analysis and numerical simulation. Learners move from vector spaces and linear maps to matrix factorisations (LU, QR, Cholesky, eigen, SVD) and apply them to projection, least squares and spectral problems. Mastery is demonstrated when a learner can choose the right factorisation for a problem, implement or specify it accurately, and reason about numerical conditioning.

Learning Outcomes

Characterise vector spaces, subspaces, bases and dimension and compute them for concrete matrices.

Compute determinants, traces, ranks, nullspaces and column spaces and relate them through the rank-nullity theorem.

Derive and apply the LU, QR, Cholesky, spectral and singular-value decompositions.

Diagonalise and spectrally analyse symmetric, Hermitian and normal matrices.

Project onto subspaces, solve least-squares problems and interpret the normal equations.

Use the SVD to compute pseudo-inverses, low-rank approximations and condition numbers.

Reason about orthogonality, Gram-Schmidt and unitary transformations in finite dimensions.

Translate applied linear problems (regression, PCA, PageRank, Markov chains) into matrix form.

Topics Covered

Vector spaces, subspaces, span, linear independence and bases

Linear maps, matrix representations and change of basis

Rank, nullspace, column space and the rank-nullity theorem

Determinants, cofactor expansion and multilinearity

Inner products, orthogonality and Gram-Schmidt

Projection matrices and least-squares via the normal equations

Eigenvalues, eigenvectors and the characteristic polynomial

Diagonalisation and the spectral theorem for symmetric matrices

LU, QR and Cholesky factorisations and their uses

Singular value decomposition and the Eckart-Young theorem

Pseudo-inverse, condition number and numerical rank

Positive-definite and positive-semidefinite matrices

Quadratic forms and definiteness tests

Linear-algebraic view of PCA, regression and Markov chains

Matrix calculus notation for gradients and Jacobians

Skills Developed

Matrix factorisation Linear-system solving Eigen-analysis Orthogonalisation Low-rank approximation Numerical conditioning assessment Translating applied problems into matrix form Working across symbolic and numerical linear algebra tools

Suggested Resources

Strang, G. (2016). Introduction to Linear Algebra, 5th ed., Wellesley-Cambridge Press

Trefethen, L.N. & Bau, D. (1997). Numerical Linear Algebra, SIAM

Axler, S. (2015). Linear Algebra Done Right, 3rd ed., Springer

Golub, G.H. & Van Loan, C.F. (2013). Matrix Computations, 4th ed., Johns Hopkins

Horn, R.A. & Johnson, C.R. (2012). Matrix Analysis, 2nd ed., Cambridge

MIT OpenCourseWare 18.06 Linear Algebra (Prof. Gilbert Strang)

3Blue1Brown — Essence of Linear Algebra video series

Assessment

Learners are judged on their ability to formulate matrix problems correctly, select factorisations and reason about conditioning. The weighting rewards transfer to applied settings over mechanical calculation.

Written problem set portfolio across five sub-topics35%

Computational project applying a factorisation to real data30%

Timed closed-book examination20%

MCQ diagnostic bank (this module's 200+ item pool)15%

Practice bank: 100 MCQs · Timed mock & self-paced modes

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