The Art of Linear Algebra [pdf](raw.githubusercontent.com) |
The Art of Linear Algebra [pdf](raw.githubusercontent.com) |
I like how the author sets up a "grammar of matrix multiplications," and then reuses the same patterns in the rest of the document.
For people who might not be familiar, these visual notes are inspired by and complement Prof. Strang's new book https://math.mit.edu/~gs/everyone/ and course https://ocw.mit.edu/resources/res-18-010-a-2020-vision-of-li... https://www.youtube.com/playlist?list=PLUl4u3cNGP61iQEFiWLE2... see also https://news.ycombinator.com/item?id=23157827
https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFit...
[1] https://www.youtube.com/watch?v=ei6RfbplYZM
What am I missing?
This article should have been titled as "Graphic Notes on Linear Algebra for Everyone", and Prof. Strang kindly suggested this big name. I was lucky that this drew this attention.
There are some other visuals I'm trying around the area. - Eigenvalues https://anagileway.com/2021/10/01/map-of-eigenvalues/
-Matrix classification https://anagileway.com/2020/09/29/matrix-world-in-linear-alg...
When I was an undergraduate, I didn't get this understanding of linear algebra... but after watching all the Prof. Strang's 18.06 classes in MIT OpenCourseWare, now I have much clear view of this area... So I really appreciate his way of teaching.
BTW, I even made a T-shirt and sent him ! https://anagileway.com/2020/06/04/prof-gilbert-strang-linear...
In undergrad, my mnemonic for these operations was visualizing the matrices animated in my head. The more complex ones, it was actually easier for me to remember Scheme functions that represent the algorithm (all expressed via higher-order functions so it was pretty concise); this was unique to my circumstances as an undergrad, not something I can pull off today without reviewing a lot of material.
Presenting the operations with color and blocks just gives a more natural "user interface" (lacking a better term) for remembering it!
Also, they say "rank 1 matrix", but they haven't defined the concept of rank yet.
Some readers might find this kind of presentation acceptable, but personally I strongly dislike it when concepts are used before they are defined.
1. Theoretical (as in vector spaces, etc). These mostly are useful in advanced courses in engineering/science, and a lot of their applications involve calculus (e.g. Fourier series, function spaces, etc). So calculus needs to be taught first.
2. Computational. These can be subdivided into applications that involve calculus (e.g. differential equations) and everything else (graphics, etc).
Many of the latter's applications are relatively recent (last few decades). Whereas calculus was needed in virtually all types of engineering and science. So it made sense to teach calculus.
Imagine it's the 1970's. Your in HS. What will you do with all the linear algebra knowledge that won't require calculus? Assume you have no access to computers.
I personally find geometry and algebra more interesting, but it seems to me that derivatives are more fundamental than matrices. But maybe that's just my bias from being educated like that.
At a high school level, you have students who are either engaged and will likely learn both subjects eventually, or students who aren’t as engaged and are just looking to take their “last math class”. In my opinion, it makes more sense to offer a choice or at least focus on the curriculum that keeps students are that age to most engaged.
As someone who has studied both subjects, I’d go with linear algebra 9 times out of 10, the exception being if someone wanted to also study physics.
Also I would add that linear algebra, calculus, and differential equations all go pretty much hand in hand. We could probably stand to teach anyone with an inclination for STEM all of those much sooner.
Possibly not Calculus 2, but at least getting an intuitive understanding of rates of change, second derivatives, and area under a curve seems pretty critical for all the sciences.
IMHO, since linear algebra is largely a tool for solving differential equations, I think calculus should be taught first, as the fundamental knowledge.
But calculus is also very very useful, and probably easier to understand, derivatives and integral are quite intuitive concepts compared to eigen vectors...
When worded this way, it sure doesn't sound like something worth doing. Matrix algebra (computational rules for matrix-vector and matrix-matrix products) is just an "implementation detail" of the general idea of a linear transformation.
The notion of a linear transformation T(x) = y where x is an input vector and y is an output vector is a really good thing to know ASAP so I'm all for the LA before CALC... or rather, if I had to choose between one XOR the other, I'd go for LA for sure!
1/ For practical considerations, the notion of a linear transformation is super useful if you'll be studying any complex process (as soon as you have multiple input variables, you'll want to put some coefficients in front of them, and what is the simplest math model you can use? In high school math we learn about proportionality relations, i.e. y = mx, where the output y depends on the input x multiplied by coefficient m (the slope if you think geometrically in the xy-plane).
Extending the notion of proportionality to transformations with n inputs and k outputs, instead of the single slope m you need k*n coefficients to describe the proportionality relations between input component j and output component i.
Linear models are pretty good in the bang-for-your-buck metric for math models since: (1) k*n params for an R^n --> R^k transformation is reasonable amount of parameters, and (2) using "tomography*" you can easily estimate each of the coefficients. This is why LTs are used in many fields of science/computing use linear transformations (Biology, Chemistry, Economics, Statistics, NNs, etc.).
tomography*: input n "probing inputs" to T e1, e2, ..., en and record the outputs T(e1), T(e2), ..., T(en) (each of the outputs is a k-dimensional vector) --- if T is a linear transformations, then the info you've collected is enough to know all the k*m coefficients of the linear transformation (<=> entries of the matrix).
2/ From a theoretical point of view, I think teaching linear transformations and matrix-vector products (the boring row-times-column arithmetic rules) is a really good thing since it introduces learners to representation theory. You have one thing in math land y = T(x) and another thing in math land y = Mx and you know their behaviour and properties are identical (isomorphic?). This means you can understand the properties of one of the things by studying the properties of the other thing, e.g., Ker(T) <=> Nullspace(M).
For me, this first contact with representation theory concepts feels like a really valuable thing to have. (A good knowledge buzz moment to get learners more interested in learning math). And it's not just linear transformations and matrices that have a "is a representation of" relationship between them. There are lots of representations in LA:
- vectors <-> coordinates
- system of equations <-> matrix equation
- row ops <-> elementary matrices
- linear transformations <-> matrix-vector products
- composition of linear transformations <-> matrix-matrix products
- graph <-> adjacency matrix
- conditional prob p(y|x) <-> matrix whose columns are p(y|x_i)
- function in time <-> Fourrier coefficients
- quantum state <-> vectors with complex coefficients
- quantum operation <-> unitary matrices
- quantum measurement <-> list of projection matrices that sum to the identity
3/ From a pedagogical point of view, if we can deliver the "representation theory buzz" from linear algebra in high school, then this will be a good chance to review some important high school ideas:
Integer representations:
- integers in terms of decimal digits with place value
a = dn...d3d2d1d0 = dn*10^n + ... d3*1000 + d2*100 + d1*10 + d0
- integers in terms of prime factorization
a = 2^a2*3^a3*5^a5*...
- fractions as m/n where m in Z and n in N\*
- reduced fractions as m/n where m in Z and n in N\* where GCD(m,n)=1
I remember when we were introduced to abstract vector spaces in high school, and we were all pretty confused, even though this was a high school dedicated to mathematics, the foremost in the country.
Even complex vectors had us scratching our heads, which in retrospect seems absurdly trivial. It's just that we were used to thinking in very concrete terms, anything purely abstract is 10 times harder to grasp, so you probably can't teach things like linear transforms without matrices.
Derivatives and integrals on the other hand are very easy to visualise.
If I were to think about what a high school student would get the most value out of, it would realistically involve a combination of linear algebra and statistics (getting into basic linear modeling/ols). You’d have to hand wave away some of the proofs which require knowledge of calculus, but high school classes aren’t very rigorous anyways.