I hope the author is reading your comments, because they make it clear that at least one of his readers is coming away with seriously incorrect ideas about linear algebra!

The thing that you are missing is that linear algebra is about much more than matrix multiplication. It is about the properties of linear transformations in general. While matrix multiplication may have simple implementations in spreadsheets, many other concepts in linear algebra don't, for example calculations of and properties of eigenvectors and the corresponding eigenvalues, or the various matrix decompositions (themselves closely related to eigenvalues and eigenvectors). The matrix decompositions in particular give results that not only are mathematically interesting, but are enormously important computationally.

In terms of societal impact, other than calculus, no field of mathematics has had a greater impact than linear algebra. From quantum mechanics to the google search engine to machine learning, linear algebra is fundamental. Spreadsheets are awesome and a brilliant idea, but they are aimed at solving a different set of problems. Important though they may be, they pale in comparison to linear algebra.

My point isn't that it lacks utility, it's that the proofs are not relevant (maybe they are to the teeniest tiniest number of people but I'm not sure of even that - most research is about speeding up calculations on computers). It's that practically it amounts to a library of computer functions. I can type the date function on the command line, how it works behind the scenes is not relevant.

The details of the proofs are perhaps not relevant to a practitioner. Although I trust you would agree that the existence of the theorems are very useful?

You seem to have the idea that the only useful thing to come out of linear algebra are a set of algorithms, and that as long as you have computer routines implementing those algorithms, the proofs that those algorithms work don't matter. There is a grain of truth in this, but the concepts that you learn in a linear algebra course - particularly around eigenvalues and eigenvectors - are very important in providing you the knowledge to choose those algorithms well. And the process of working hard to understand proofs is a great way (perhaps the best way) to make sure that you really understand those concepts. It's also a great way to make sure that you really understand what a theorem means / why a particular algorithm works.

most research is about speeding up calculations on computers

A a large amount of progress in "speeding up calculations" comes from algorithmic advances. These advances aren't being made by people who view linear algebra as nothing more than a set of library routines. They come from people who deeply understand the concepts and proofs behind the important theorems of linear algebra.

"most research is about speeding up calculations on computers"

So, one of the most convenient things that Linear Algebra does is about making things computationally cheap. Without spending too much time since I can't tell if you're a troll :), this branch of mathematics often times provides the easiest way to optimize data of a certain characteristic. People want to save money/time/labor/etc., and a lot of times linear algebra provides the best framework for doing so, and when you have numbers and data on a large scale it often times becomes infeasible to fiddle around with individual cells in a spreadsheet.

RE: relevancy of proofs -- I think a lot of the most interesting and valuable research in this subject is about figuring out mathematical mappings/equivalence of simple LA function to replace laborious procedures in differential calculus and what not. I personally hated proofs when I went through the education system myself, but the value of doing them is to get an abstract and less-biased understanding of things over learning metaphors that may not necessarily apply precisely to any given situation or problem.

(Disclosure: I'm an operations research and math guy doing data science, so bias here.)