Linear Algebra - MAT00050I

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Module summary

This module covers the basics of concrete, computational, and abstract linear algebra, motivated by the study of solution sets to systems of linear equations, linear transformations and their invariant subspaces, and inner product spaces.

Related modules

Module will run

Occurrence	Teaching period
A	Semester 2 2025-26

Module aims

This module covers the basics of concrete, computational, and abstract linear algebra, motivated by the study of solution sets to systems of linear equations, linear transformations and their invariant subspaces, and inner product spaces.

Module learning outcomes

By the end of the modules, students should be able to:

1. Work with the concept of a linear subspace;

2. Use the notion of dimension, and calculate dimension using the concept of a basis;

3. Describe and recognise linear maps, and use key concepts such as kernel, image, and the Rank-Nullity theorem;

4. Calculate eigenvalues and eigenvectors of linear maps;

5. Define and work with real and Hermitian inner products and basic consequences of orthogonality;

6. Carry out matrix factorisation algorithms using a computer algebra system.

Module content

• Linear systems of equations in n real or complex variables and their solutions: linear combinations, linear span, subspaces, independence and bases, dimension. Intersections and sums (direct sums) of subspaces and their relationship to systems of equations.

• Linear transformations on coordinate space: linearity of maps between coordinate spaces, composition as matrix multiplication, rank and nullity, one-to-one and onto as consequences of nullity and rank. The Rank-Nullity Theorem and its relationship to existence and uniqueness of solutions to linear systems and the dimension of solution spaces.

• Eigenspaces, diagonalizability, necessary and sufficient conditions for diagonalizability. The characteristic polynomial and the Cayley-Hamilton Theorem. Minimal polynomial. Invariant subspaces and the idea of Jordan form (w/o proof).

• Abstract linear algebra: the axioms of a vector space. All previous ideas translated to this generality. Coordinates via bases and coordinate representations of linear maps; change of coordinates. Dual spaces and dual maps.

• Real and complex inner product spaces. Orthogonality, orthogonal/unitary bases, orthogonal projections. Gram-Schmidt process and QR factorization. (P)LU factorisation and Crout’s algorithm with and without pivoting.

• The adjoint of a linear map. Self-adjoint maps and consequences for eigenvalues and eigenspaces.

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy

Indicative reading

To be added