All UCL Maths Year 1 Modules Ranked

I’ve just completed my first year at UCL studying Mathematics, and am quite pleased to be averaging a 91%, with a mark above 90% across six modules :D

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

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

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 R^n and 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 allows us to compute 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 found myself to be really 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 you encounter 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 I had to apply the row-reduction algorithm like five times in an exam. To that end, 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 but also felt like it wasn’t the best way to design the course like this.

Tier and explanation

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

TESTING

The \(\epsilon\)-\(\delta\) Definition of a Limit

One of the foundational concepts in real analysis is the formal definition of a limit. We say that the limit of a function \(f(x)\) as \(x\) approaches \(c\) is \(L\), written as: