NB: I think many of these linear algebra proofs would benefit (in terms of legibility) if the dimensions of the matrices/equations were annotated beneath them. (I created a LaTeX macro for my master's thesis to do just that.)
These are what are known as prerequisites. This expectation on HN that all science and math should be made approachable without requisite prior knowledge is so bizarre to me. On the other hand, if blog post is about move semantics, nobody chimes in "ha, try understanding that without knowing about constness first".
I doubt anyone expects this, but there are sharp trade offs involved. The more accessible you make a presentation, the wider the possible audience. With each year of additional technical training you expect, the total audience drops significantly; a math article targeting people whose math training only made it to an introductory calculus course can be read by perhaps 1000 times as many people as a math article with prerequisites from grad-school pure math coursework.
\underset{m,r}{U}\underset{r,r}{\Sigma} = \underset{m,n}{A}\underset{n,r}{V}
\ddim Umr \cdot \ddim {\Sigma}rr = \ddim Amn \cdot \ddim Vnr
\newcommand{\ddim}[3]{\ensuremath{\underset{\scriptscriptstyle #2\times #3}{\rule[-1ex]{0ex}{+1ex}#1}}}
Looks like this:
PS: I'm not sure if it's something that bothers other people but sometimes it bothers me. Especially notable that to me there are some domains that seem to use weird notation as a shibboleth or to make it artificially harder to think about.
There are plenty of other reasons mathematical notation might be hard for beginners:
* The notation in question might have originally been invented for a narrower purpose than it's being used for now, and has been stretched beyond the domain where it fits well.
* Someone made a weird decision a hundred years ago, but now everyone's used to writing things that way, and if you want to be understood it's easier to use the same notation as everyone else.
* The notation might actually be better than you're giving it credit for, but this is only obvious once you've spent a while working in the field.
Overall, I've noticed that people coming to math from CS tend to put more weight on the question of notation than mathematicians do. I will say, from my perspective, I think it's very easy to make a bigger deal out of notation than necessary. A lot of math is just authentically pretty tricky and takes effort to learn no matter what notation you use, and I think it's pretty rare for the notation to actually be the lowest-hanging fruit here. This isn't to say that there's no mathematical notation that could use improvement (quite the opposite!) but I don't think making those changes would have quite the impact you might be imagining.
* The tendency of math to have many different notations to express the same thing. My calculus textbook started by apologizing for the like seven different notations for "take the derivative of this function". Would it kill mathematicians to just pick one notation and standardize on it?
* The number of times the same notation can mean two different things. I've always been particularly bothered by stuff like implicit adjacency is multiplication, except when it's not. Is "a(b + c)" supposed to mean "add b to c and use the result as the argument for a" or is it supposed to mean "add b to c and multiply it with a"? You'd think it should be obvious, but functions are a ring, with multiplication corresponding to composition...
I do agree with you that calculus textbooks often go overboard with notations for the derivative, and in fact I remember being taught some (like a capital D with a subscript x) that I then never saw again for the rest of my life.
But as to the question of why "mathematicians" would do this, I don't think the answer is all that mysterious: "mathematicians" is a category that includes a lot of different people in different places and there's no central committee that enforces notation standards! It's like that one XKCD comic everyone posts about inventing new standards.
\begin{equation*}
\begin{bmatrix}
a & b & c \\
d & e & f \\
h & h & j \\
\end{bmatrix}_{3 \times 3} = 5
\end{equation*}