Starting again at 58. Scary, right? And a little bit mad.

I’m preparing for my PhD qualifying exam and deep in a literature review, so this Saturday morning finds me doing something I didn’t expect: a full refresher in linear algebra.

To my surprise, the challenge isn’t a “rusty brain.” It’s the shift in perspective.

In quantum physics, linear algebra stops being about formulas and becomes geometry.

States as vectors. Measurements as projections. Transformations as rotations in spaces you can’t see but must somehow feel.

It’s a very different mindset from my years at Barclays Global Investors (BGI), managing one of the Aquila quant funds. Back then, mathematics was a tool for taming uncertainty. Everything was grounded in data, estimation, and the classical assumption that the world behaves if you model it hard enough.

Quantum physics flips that intuition. Instead of forcing the world into a model, you start by accepting that the world is a model – a mathematical structure evolving according to rules that don’t care about your priors.

In finance, I used linear algebra to extract structure from noisy markets. In quantum physics, I’m learning to see structure as the thing that creates the world in the first place.

So here I am, back at school decades later, rediscovering the foundations with a completely different mind. I wondered why the suggested first class is linear algebra, followed by quantum mechanics. But it makes sense because quantum theory is fundamentally expressed in that language:

• Quantum states are vectors in complex Hilbert spaces.
• Observables are operators acting on those vectors.
• Measurements correspond to eigenvalues and eigenvectors.
• Entanglement and superposition are geometric relationships between subspaces.
• Unitary transformations describe time evolution and symmetry — all matrix operations.

Starting with linear algebra ensures you can think geometrically about quantum systems rather than just manipulate formulas. It’s the grammar of quantum mechanics — without it, the rest of the theory is unreadable.

I’m also hoping to take my inspired 16-year-old godson on this journey – his journey – into quantum physics. You can read about that in Arthur & Toby.

And yes, the screengrab from my MIT online course shows the researchers using paper and pen, just like me:

I thought I was a dinosaur. But mind-mapping really does help you see the whole picture – and make the connections that matter. It definitely helps you to think in a non-linear way, like neural networks!