All UCL Maths Year 1 Modules Ranked

Sun, 28 Jun 2026 00:00:00 +0000

I’ve just completed my first year at UCL studying Mathematics, and am quite pleased to be averaging a 91% :D

I thought it’d be nice to reflect a bit on the modules, so I’ll be going through each module one by one, and putting them into a tier.

This post is intended for anyone interested in pursuing undergraduate mathematics, perhaps at UCL, or anyone with a university-level mathematics background.

MATH0005 Algebra 1

Module content

This course is an introduction to linear algebra, extending the basic matrix and vector concepts introduced in A-Level (or equivalent). The module begins with the basics of matrix arithmetic (multiplication, inversion, transposition) before introducing the row reduced echelon form (RREF), which is a nice way to solve large systems of linear equations, and determine if a system has a unique solution, infinitely many, or none at all.

Then, the module looks into vector spaces \(\mathbb{R}^n\) and \(\mathbb{C}^n\), where a bunch of fundamental mathematical vocabulary is introduced: linear independence, spanning sequences, bases, dimension etc. With all that machinery developed, the “most important theorem introduced in first year” can be proven: the Rank-Nullity Theorem, which informally, is a “conservation of dimension” law for matrices.

From there, we learn that matrices can also be interpreted as “linear maps”, actions that stretch or rotate space. This leads into the chapter where we want to find “eigenvalues” and “eigenvectors” of matrix. Eigenvectors are the special arrows that stay in that direction when we apply the matrix transformation, only stretched by a specific scale factor – the eigenvalue. This is useful in the process of “diagonalising” a matrix, which is helpful for computing large powers of matrices easily with a closed-form formula.

The final chapter defines lengths and angles in multi-dimensional spaces. We use the Gram-Schmidt procedure to construct perpendicular “orthonormal” bases, and then prove the Spectral Theorem for symmetric matrices.

My thoughts on this module

Personally, I found this module to be really challenging the first couple weeks and was quite lost. But by exam season, I actually found myself to be very confident in this module for two reasons. One - the exam actually was extremely computational (more on this later). Two - I gradually got used to the abstraction encountered in university Mathematics throughout first year.

The exam largely focused on computing things, where you can already secure most marks by learning the essential algorithms for each chapter (solve a system of linear equations via RREF, find a basis for the kernel/column space of a matrix, apply Gram-Schmidt, compute determinants, diagonalise a matrix). I guess this made it easy to score a high number of marks, but it also made the exam really boring where it was just applying the row-reduction algorithm like five times in an exam. So I guess I wish the exam was more proof-based.

It seemed like there was a large burden of proofs initially, but it turned out that we did not need to learn 90% of the proofs to the theorems we learned in lectures, which I appreciated to a certain extent.

Tier and explanation

B tier. I thought it was a nice, standard intro to linear algebra.

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All UCL Maths Year 1 Modules Ranked

MATH0005 Algebra 1

Module content

My thoughts on this module

Tier and explanation