An Intuitive Guide to Linear Algebra

Despite two linear algebra classes, my knowledge consisted of "Matrices, determinants, eigen something something".

This is a companion discussion topic for the original entry at

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@Tom Elovi Spruce,
I believe, the topic of the article at hand is math, as opposed to literature, but if it were, you’d have a valid point.

Great choice of topic! I jumped on this one hoping to refresh my memory on linear algebra and reconsider its usefulness, but in this article it was a bit of a bumpy ride. I would say your usual style allows for a much smoother transition from building blocks to a-ha moments.

In “Organizing Inputs and Operations”, if you look carefully, this will read smoothly to someone who already understands what you’re talking about, but a novice would be lost. You introduce two different operations at the same time as you’re explaining what the rows mean in the matrix notation, leading to both points being hard to catch. Going forward, you frequently forget you’ve not introduced a notion before you start using it (“transformation”), using the “axes” in the globe analogy without really explaining what you’re doing there etc.

I hope you take it well - this article definitely is better explained than what I got in college, but I came to expect even better from you :slight_smile:

“No! Grammar is not the focus.”

To be honest, that part of the article threw me off because it sounded like you were criticizing yourself for shifting the focus from math to english.

I think you jumped to the analogy too abruptly and its link to how matrices are taught isn’t clear.

[…] An Intuitive Guide to Linear Algebra — Here’s the linear algebra introduction I wish I had. I wish I’d had it, too. (via Hacker News) […]

I have to disagree on the “spreadsheet” approach to linear algebra. Matrix/vector multiplication never made any sense to me, until I realized it’s just projecting the vector onto the original identity basis, and then reconstituting it using the new basis instead. You can discover and draw this process entirely visually.

The relationship between a matrix and the vectors made up of its rows or columns is ridiculously obvious once you see it in action. Yet in years of linear algebra and engineering, nobody ever bothered to show this to me.

A spreadsheet is actually a much more general structure than a vector space, but leaving that aside, there are two insights being mashed into one here:

  1. Many problems, though not obviously geometric, can be shoved into some geometric form and more easily tackled thus.
  2. There is a scatter of algebraic structures describing features of various geometries that have proved useful over the years. The scatter is roughly a cone from the origin of unstructured sets out to things like noncommutative geometry.

There are other things near to vector spaces in that scatter, like affine spaces (vector spaces without origin) and modules (vector spaces over rings instead of fields), and all kinds of fascinating things that show up (look up tropical semirings and you’ll go down an enormous rabbit hole, all without ever leaving familiar algebraic operations). Reach a little farther out and you find yourself with inner product spaces, then add some calculus and you get Banach spaces, and a little farther in Hilbert space (weird fact: there’s only one Hilbert space; all its various appearances are all isomorphic). Keep going and you start finding yourself in Riemannian geometry, and then even farther out in noncommutative geometry and the like.

Vector spaces are a sweet spot, for three reasons:

  1. They’re sufficiently unstructured where most of the components of more complicated geometries like tangent spaces and the like will all be vector spaces.
  2. They’re just enough structure to always be able to express any abstract vector space in familiar vectors of numbers and operations on them as matrices. Thus you can always grab a basis and start computing, no matter how exotic your vector space may seem.
  3. They’re turtles all the way down. The space of linear operations on a vector space is a vector space. The space of coordinate transformations from one basis on a vector space to another basis is a vector space. When you start adding inner products and the like, you can pretty well always find a way of looking at them where they’re just another vector space.

They have oddly nice properties as well. For example, no matter how weird the vector space, it has a well defined dimension (though it may be infinite).

Linear algebra “done right” is really a question about the structure that emerges from a very broad class of geometric problems. The really interesting part is how you suture vector spaces together in various ways to get other classes of geometries entirely. For instance, a curved space isn’t a vector space, but define a tangent space at every point of that space. The tangent space at a point is a vector space with the same dimension as the space. You can think of it as the velocity of an object at that point. The geometry comes in when you use the notion of tangents as velocities to map back to actual paths in the space.

So matrices and vectors of numbers are nice, but they’re barely the tip of the ice berg of linear algebra.

@rrdillon Glad it clicked!

@Mladen: Thanks for the feedback! Yes, the “Organizing Inputs and Operations” section is the tricky transition, I’ll have to see if I can make it a bit smoother. One thing I love about the net :).

Thanks Tysen, just fixed!

Thanks mophism, appreciate the suggestion!

thank you very much!)

[…] is in reply to this linear algebra guide, which came up on Hackernews today. The author wrote that he didn't have a good intuition about […]

I wrote some of what I know about eigenvalues and determinants down here:

Maybe it will be helpful for intuition.

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When I printed this, the first letter of every line was cut off. Can you fix the print css?

Like most things the best way of learning something is to approach it from different viewpoints. This article does that although not convinced about keeping examples abstract.

I only really understood the advantages of a Matrix when I had to write a program to rotate points in 3d space on a different course. Did the rotation equations then the whole concept of matrix maths ‘clicked’ when I realised its nothing special it’s just a neat way of doing the maths!

In fact the whole mystery of maths clicked - maths is nothing more than a human language and tool for describing how things interact. Maths doesn’t have rules it simply implements observations of physical reality in a convenient way. For example complex numbers in Electrical engineering integration etc.

Of course the discovered rules can then hint at other physical rules that haven’t yet been discovered, which is the real power of maths in general!

I think you’re glazing over the main point of matrices:

     Every linear map can be represented by a matrix.

This should not be obvious to the beginning student. We don’t work with matrices just because they give us a useful way to organize information, because that simply wouldn’t be useful if we couldn’t use them to represent any linear map. We work with matrices because they completely characterize the functions we care about.