alt Hacker NewsWell, he's one of the most famous and best-respected educators of linear algebra, and this is an unusually basic piece of linear algebra to be taking aim at him for, so one answer is: if you have to ask whether his approach is defective, you should first evaluate how strong your own understanding is. That's argument from authority, but then: Strang is an authority. The typical push-pull on a message board is between Strang and Axler, and, if you want to find out if Axler is going to save your argument, flip to 6.A in LADR.
The direct response to your question is: the centrality of the angular interpretation of the inner product is the kind of thing I feel like you'd say if your primary purpose for learning linear algebra is to program video games. Linear algebra isn't "about" geometry, and, in particular, Strang's teaching goal centers vector spaces and the relationship between spaces. You need inner products to apply linear transformations with matrices. You basically don't need the angular interpretation... at all? if your goal is to teach linear algebra as a tool for data manipulation and, as Strang did in his last course, as a vehicle for deep learning.
(You need orthogonality, but you didn't say `a⟂b=0`, you said `|a||b|cosθ`).
I'm not rating this blog post "higher" than 3B1B; the comparison doesn't even make sense. The blog post and the video series simply have different objectives.
Replies
> That's argument from authority, but then: Strang is an authority.
Yes, it's an argument from authority.
> The typical push-pull on a message board is between Strang and Axler
We're not playing Pokemon with linear algebra textbooks here...
> the centrality of the angular interpretation
It's not central. It's one of the ways to think of it. I think 3B1B actually did a good job emphasizing this in his video series: there are many ways to look at vectors, and all of them are sometimes useful. They can be arrows in space, or sequences of numbers, or black boxes that obey the vector space axioms, or polynomials, and so on.
> if your primary purpose for learning linear algebra is to program video games.
What an odd guess. Seriously, I would be surprised if most math teachers didn't mention angles and norms, scalar projections, etc. An important part of math is being able to see things from multiple angles (no pun intended), and this is a useful angle to view the dot product from.
> Linear algebra isn't "about" geometry
Sort of. Linear algebra proper, the study of vector spaces and linear maps between them, is not about geometry, but geometry comes in almost precisely once you equip the vector space with an inner product, such as the dot product. A bare vector space has almost no geometric content, but the inner product gives you lengths, angles, isometries, orthogonality, and all that jazz.
> Strang's teaching goal centers vector spaces and the relationship between spaces.
That's a good goal.
> You need inner products to apply linear transformations with matrices.
I think you have a fundamental misunderstanding of linear algebra here. You do not need an inner product to "apply linear transformations with matrices".
> You basically don't need the angular interpretation... at all? if your goal is to teach linear algebra as a tool for data manipulation
Off the top of my head, cosine similarity? It's not uncommonly used.
But if you're teaching linear algebra for data manipulation, you rarely need the dot product, either. Most "dot products" in data manipulation, like the ones in this article, would be better expressed as row-vector-column-vector matrix products.
> (You need orthogonality, but you didn't say `a⟂b=0`, you said `|a||b|cosθ`).
I said "||a||·||b||·cos(θ) or at least talks about how the dot product relates to parallel and orthogonal vectors". Anyway, it's hard for a person not to interpret orthogonality as a statement about angles, so I'm not sure what distinction you're trying to draw here.
> I'm not rating this blog post "higher" than 3B1B
I guess I misinterpreted "its sauce is stronger", then.
Sorry for any mistakes; this took way too long to type.
You don't need inner products for linear transformations. You just need the idea of a basis and linearity. You define your transformation on a basis (which is all a matrix is: the list of where the map sends each basis element), and it is automatically defined everywhere else via linearity. The textbook my undergrad class used (Curtis) doesn't define inner products until after linear transformations and matrices, for example.
The angular interpretation and geometry are basically the entire point of inner products (inner products are how you define a large chunk of geometry). Angles and projections are the entire intuition behind talking about orthogonality, which is super important practically to basically every field.