Visualising Linear Algebra

I was recently in a physics education research meeting where somebody commented that it was impossible to visualise abstract linear algebra, or words to that effect. I objected that I do visualise linear algebra, and being the only one of this opinion (probably also the only one who would argue that maths is made of the same stuff as poetry) I was asked to explain. I stumbled through a few unconvincing sentences before giving up, but I thought I’d try it again now that I’m not so much on the spot. Of course, putting things into words both adds to and takes away from the idea in my head, but perhaps this will be interesting, if not useful.

Floating in a void of blackness is a point of cyan light. The colour is not terribly important, but it does identify the vector – this point of light is a vector – as belonging to a particular vector space. A vector from another space might look nearly the same, but it would be red, or green, or some other colour. Now that I’ve identified the void as a vector space, I realise that of course it isn’t empty: there’s not just one, but many, infinitely many, points of light floating there. (This reminds me that finite vector spaces are probably a thing, although not of interest to most physicists, and irrelevant right now. There are infinitely many vectors in our vector space.)

It would be impossible to draw a picture of this: infinitely many points of light blurring together so that they fill all of space, but entirely distinct, each one contained within itself. Fortunately the visual cortex appears to be somewhat more flexible than the retina. Since this is a vector space, and not a less orderly algebra, the points of light are neatly lined up in rows and columns – up, down, right, left, backward, forward, and more directions that are hard to describe. Most interesting vector spaces have more than three dimensions. I can usually only see three or almost four of them, but I have the uncomfortable sensation that one or two or infinity more are lurking behind me, watching. They’re smug, not malicious, but as an undergraduate student I had at least one nightmare about vector spaces.

Enough about the vector space as a whole – what happens if we look at an individual vector? Well, technically we can’t see any detail just yet. We need to choose a basis that pins the dimensions into place so they can’t lurk behind me or spin around each other just for the fun of it. But snapping a basis into place, we can zoom in on one of the distinct-but-continuous points of light. It looks a little like a planet covered in skyscrapers, or like the output graph from a spherical LHC calorimeter. There’s one skyscraper for every dimension; I can’t get all the skyscrapers into focus at the same time, because I’ve decided that this vector space is a Hilbert space, and has infinite dimensions. I can move along the vector space in any direction, and watch the skyscrapers change. Just one skyscraper changes with each hop, becoming a little taller or a little shorter.

While we’re looking at this vector, we can pull in another one, with an entirely different (or, for that matter, identical) skyscraper pattern. Smushing them together is the process of vector addition. Some skyscrapers reinforce and grow taller. Others cancel each other out, because skyscrapers can be either positive or negative – dark or light – and while I often ignore the distinction, it matters in this case. As the new vector shimmers into existence there’s a gentle pop and we teleport to a new location in the vector space.

I’ve taken a fancy to this vector, so I decide to change the basis to make it look special. As the new basis slides into place, the skyscrapers here all merge into one incredibly tall tower. It’s easy to pull in other vectors and compare them, now: how prominently does this megatower feature in their landscapes? Technically, I didn’t need to change the basis to calculate these inner products, but this is far more convenient than comparing each of the infinite skyscrapers on each vector.

When I drop the constraint of the basis, I’m forced back out into the sea of twinkling cyan that makes up the vector space at large. But the relationships I found before still hold, although it’s hard to see that here. I can start to find patterns, like golden threads, that flex with the vector space as the dimensions shift and change, but remain as markers of a complicated kind of order I couldn’t see before.