We will cover the basics of linear algebra, including proofs. This will start from the perspective of vector spaces over arbitrary fields, quickly specialising to real and complex vector spaces. We will study linear maps between vector spaces, how these can be realised as matrices, and how this can be applied to solving systems of linear equations.

Alternating year course, first taught AY2021 (and again in AY2023, etc.)

Fields, vector spaces, and bases.

Matrix operations and solving systems of linear equations.

Row reduction and determinants.

Change of coordinates.

Eigenvalues, eigenvectors, diagonalisation.

Gram-Schmidt orthonormalisation.

Familiarity with real and complex numbers will be assumed. Ideally, students will have had some previous exposure to mathematical proofs, though this is not strictly required.