An Intuitive Guide to Linear Algebra
betterexplained.com
betterexplained.com
It works, but there's no connection between that process and the intuition of a linear transformation; it's just a rote computation. And checking a long string of matrix multiplications to see if they intuitively make sense (shouldn't everything intuitively make sense?) is especially aggravating when you constantly have to interrupt your intuition to switch to a rote calculation.
I never thought to think of the columns of B as vectors that physically travel through A; to think of a dataflow or pipeline from right to left on the page. Sure, it's not a cure-all, but it'll be a useful mental tool to have.
Oh, and it's also an excellent introduction to the subject, although the Linear Operations section gets a bit muddled... first something's not a linear operation, and then it is, wat? Still, an excellent post.
Okay, that lets you visualize it (in the finite case) but it's a terrible way to sell it. Spreadsheets are booooring. Did you know that functions are vectors? Okay, better. Did you know that quantum mechanics is all about linear algebra? Okay, sold!
1. Almost any time you work in more than one dimension you will want linear algebra in your toolbox. There are a zillion methods for solving (non-linear) equations out there, and in more than one dimension, they use linear algebra. Newton's method? Incredibly useful in practice due (quadratic convergence rocks!), and with some linear algebra sauce BOOM you have Newton's method in as many dimensions as you can sneeze at.
2. Oh, by the way... did you know that the Fourier transform is linear?
3. Back to quantum mechanics... there's a thing you can do with a linear operator (a matrix is a kind of linear operator) where you get the "spectrum" of the linear operator. It's useful for making sense of big matrices. But in QM, the wavefunctions for electrons are described as eigenfunctions of a linear operator, and taking the "spectrum" of the linear operator gives you the actual spectrum of light that the chemical under study emits. Hence the name, "spectral theorem". It may be linear algebra on paper, but it's laser beams and semiconductors in the real world.
4. Oh hey, want to learn about infinite-dimensional vector spaces? Maybe some other time..
5. It's hella useful for modeling. Any model is wrong, but Markov processes are useful. Say you run an agency that rents out moving vans, and you have facilities in 30 cities. Vans rented in city A have a 10% chance of being dropped off in city B, 7% in city C, 9.2% in city D, etc. At this rate, how long till you run out of vans in city F? It's a differential equations problem with like 30 different equations! Or you could rewrite it as a single equation with matrices. You'll end up with weird things like 'e^(A*t)' where A is a matrix, and you thought "no way I can exponentiate to the power of a matrix" but spectral decomposition is like "yes way!" and you can solve the equation by diagonalization. Radical! (Basically, linear algebra rescues differential equations from the pits of intractability. I'm using rental vans as an example, but it could be a chemical reaction or a nuclear reaction or a million subway riders or whatever you want.)
So the question is:
Do you find economics, quantum physics, chemistry, engineering, classical mechanics, machine learning, statistics, etc. useful?
Then get some linear algebra in you!
- you've mixed in with nonlinear phenomena (most physical processes) with linear, and there's less and less reason to pretend linearity with increasing computer speeds (although in fairness you alluded to models being wrong)
- poo pooed spreadsheets without explaining why (you alluded to QM etc. but they don't care about useless proofs in linear algebra unless they're pencil pushing time wasters)
- implied infinite-dimensions is practically useful when it isn't (unless you're a mathematician/theoretical-somethingist seeking to extract tax-payer money)
There's little need to get linear algebra in you unless you want to waste time.
1. Yes, I mixed in linear and non-linear processes. My point is that you use linear algebra even when working with non-linear processes.
2. Yes, I poo-pooed spreadsheets as a bad way to market linear algebra (I thought that was clear?)
3. Yes, I implied infinite dimensions is useful. Linear algebra in QM tends to be infinite-dimensional.
I'm reading a lot of hostility towards mathematics in general and linear algebra specifically (you've put a bunch of toxic comments all over this thread), and I'm not really sure why.
You don't have to appeal to QM for their usefulness. Function spaces are important in applied and computational mathematics because of their use in understanding integral equations (the theory of integral equations before Hilbert spaces was a giant mess), partial differential equations, calculus of variations, approximation theory, etc. No doubt marshallp will respond that this is all bullshit because it ultimately boils to finite processes running on finite state machines, where there are no infinite sets in sight, let alone infinite dimensions. The equivalent approach to physics would be an extreme form of empiricism, banning the use of concepts like electrons (as some logical positivists actually proposed to do in the early twentieth century) and requiring all physical laws to be stated in terms of directly observable phenomena, whatever that means.
Those people may exist. But I don't see any of them here.
You've gone too far in the other direction by saying it's a stupid and complete waste of time to study math. There are real breakthroughs in insight there, even if sometimes obscured by pompous nonsense.
Your comments are fueled by anger, rather than a sincere effort to inform others, and this is why you've been downvoted.
I'm willing to bet you can be a constructive contributor here if you try. You're probably right that a simplified and more pragmatic approach to learning math might make it accessible to a larger group of people. If so, it would be better to go help make that happen than to blindly criticize everyone else.
There's a huge, huge range of frauds being committed in society. I think people should point them out when they see it.
edit: I also agree mostly with the hard-headed engineers you brought up. Physics is somewhat useful, but it's also vastly overtaught and overfunded relative to it's practical utility. I'm an AI guy and think we'll get the singularity before any of the fundamental research going on pays off (and that's only a small proportion of the world's wealth being spent on physics education for people who will never find practical use for it). And I think there's also a lot of mathemagic symbol throwing in there as well to extract tax-payer money (a lot of it caused by the mathemagicians's influence on physicists).
Also, don't feed the troll.
For me it is too computational. I prefer understanding the topic from first principles as described at http://news.ycombinator.com/item?id=4086325. (Then again I don't particularly like spreadsheets either.)
After going through that course I finally understood things like eigenvectors, null spaces, and projections. Now I see them everywhere (unless you think that's a curse)
Linear algebra is so far-reaching, I find it surprising that other branches of mathematics often seem to get preferential treatment (usually normal algebra and geometry), in spite of the fact that linear algebra is both:
a) fairly advanced (i.e. not often taught in school, at least not the deeper stuff)
b) not very difficult to learn (unlike lots of other 'introductory' topics in mathematics).
Perhaps there is something about matrices (being mere tables of numbers for most folks) that people find unattractive, almost statistics-like.
(On the other hand, it could be a simple extension of the symbol barrier [1], given those long vertical brackets.)
[1] Prof Keith Devlin introduces this concept here: http://profkeithdevlin.org/2012/02/22/how-to-design-video-ga...
Even then, it was a question I had to ask my brilliant, constantly-pissed looking young professor. "Hey, uh, the Jacobian... what does the determinant mean, uh, geometrically?". He looked at me like a slug, before explaining it was the measure of the newly mapped unit square. Fireworks went off in my head. Two linear algebra classes before only ever explained it by its algorithm or its usefulness (e.g., ∃ A^(-1) for A \in R^{n,n} iff det(A) != 0)
Side note: that professor had the most effective teaching style for pure math I've ever seen. Besides lectures that expanded on the contents of Rudin and interesting problem sets, he gave us a list of a hundred theorems, propositions, and exercises. Told us the final exam would be six problems, four of which would come from that list, another of which would be a clever new one, the last something truly hard.
Never learned analysis better than when sitting down and working through (not memorizing) each of those proofs and theorems for possible later recapitulation.
FWIW, that book has two explanations: the first on p 296 is a lot like "that's how it works out" and the second on p 320 is geometric.
> why does the determinant work to solve linear equations (i.e. Cramer's rule)?
Does the explanation on p 331 of joshua.smcvt.edu/linearalgebra/book.pdf help? (It uses the geometric understanding of the determinant.)
Recently, learning to do Principal Components Analysis to solve a handwriting recognition problem is what finally shed an enchanted light on linear algebra. The way that a simple matrix of data samples is transformed into an ordered set of principle components (eigenvectors, ordered by eigenvalue) is.. "unreasonably effective" (as they say). The principal components are your signal, and the rest (with eigenvalues ~0) are the noise. the handwriting recognition, btw, works fantastic for my simple application. no need for non-linear kernels and whatnot.
Even better now, in this hacking life after pure math in college and grad school, is that I can build intuition now not just by proofs and exercises, but also efficient, coded implementation. Gives a different feel for the tools and concepts.
As mathematicians, we didn't do any of this matrix and vector stuff with numbers when introducing linear algebra in university. There were a bunch of axioms, and you proved things. That's how you know.
What the author sees as abstract "(2d vectors! 3d vectors!)" was way more applied than the stuff we dealt in.
But, granted, the purpose wasn't learning about how to get mini-spreadsheets for equations. It was about how to rigorously navigate a useful axiomatic setting.
(Later on, we proved that you can find a base, and write down your linear transformation as a bunch of numbers and call that a matrix; same for points and vectors. But we always saw that as somewhat ugly, and anyway limited to the finite dimensional case.)
edit: downvoters downvote instead of providing refutation. A lot in common with fundamental religionists.
Theorems are useful in all sorts of unexpected ways. At the turn of the 20th century group theory was considered a worthless corner of mathematics, but understanding group theory allows modern chemical instrumentation to work.
edit:
> It seems like it's the full employment act for pencil pushers
> downvoters downvote instead of providing refutation. A lot in common with fundamental religionists
You've proven the downvoters right by piling one ad hominem attack upon another.
Let's say you need a communications channel with a 300 Mbit/s capacity. Shannon's theorem lets you know what kind of parameters make that possible -- namely, bandwidth and signal-to-noise ratio. From there, I can make an informed decision about the entire signal chain. Once I choose the bandwidth, the theorem tells me the SNR so I can assign a noise budget to each of the components. If we assume that the noise is Gaussian, we can calculate total noise level as the root sum of squares of the noise levels of the individual components (don't forget to multiply by the gain). I can look at an amplifier IC's spec sheet and immediately say, "that's too noisy" or "that's overdesigned, I think I can get something cheaper".
And thanks to theorems we got from the field of real analysis and statistics, we know that the sum of Gaussians is itself Gaussian. So rather than sticking two components together and measuring the noise, or running a computer simulation, I can simply square, sum, and square root.
Each of these theorems reduces the amount of work necessary -- whether by pen and paper or by computer -- by an enormous factor. But if you can describe in similar detail how the computer program would work, I'll acknowledge your greatness.
Really? What happens when you need to do a similar calculation across hundreds of millions of data points?
And you can't just "write a computer program that tries different parameters". You have to prove that your numerical method solves a certain class of equations first, otherwise your rocket will fly sideways, if at all.
I assume that all those automatic methods you mention are passed down from above in some sort of holy scriptures that we're supposed to blindly believe and use? Or maybe some "pencil pushing" mathematician came up with them first and _proved_ that they actually work?
It does not seem to run extra slow for any input, so it probably won't run slowly on any input in actual use. Besides, even if it did I would have no way of proving what types of inputs it failed on, or if another algorithm would do better.
Keeping the above in mind, does it actually justify the billions of man hours spent on mathematics.
Exhibit 2: In your profile, you describe yourself as "Machine Learning Expert"
Hmm.
Would you describe him as a "pencil pushing fraud" too?
It's easy to dismiss what you don't understand but you should consider the possibility that it is significantly more difficult develop algorithms like those in VW and "bang out" implementations of them that are fast and correct.
Secondly, no, John didn't say learning theory is "useful only in a crude way", he said, "learning theory is most useful in it’s crudest forms". Big difference. And besides, he says right at the beginning of the post that he believes "learning theory has much more substantial applications [than generating papers]".
To be convincing, theory needs to be precise – if you are not careful about what you are talking about it is easy to believe things that are not true. However, what I think John is saying is that value of theory comes from afterwards abstracting away the precision and understanding the main message (i.e., the "crude form") of a theoretical result. In general, I don't think it is possible to get to a convincing "main message" or "crude form" without someone having grappled with the details.
No matter how many experiments you run, you will only ever show that an algorithm works well or not in a typically small finite number of cases. What theory does is look those cases and ask something like, "It seems that every time X is true of my problem, an algorithm with property Y works really well. I wonder if that is always true?" This type of question gets carefully formalised and then (hopefully) answered. The process of formalisation (i.e., defining things carefully) can yield new ways of thinking about things (e.g., over fitting and bias-variance trade-offs), and having an answer to the general question means that you can be assured that future uses of your algorithm will behave as expected.
You seem to have an unshakeable opinion that mathematics/theory/"pencil-pushing" is inherently a waste of time. That's a real shame. Why do you believe the pursuit of precision, insight, and proof are somehow inferior to running experiments? I find both to be valuable and the interplay between the two extremely rewarding.
To avoid any fluff, let's go with a concrete example: some of the best performing algorithms for segmentation are based on spectral clustering (http://en.wikipedia.org/wiki/Spectral_clustering). What does an eigenvector corresponding to the second-smallest eigenvalue of the normalized graph laplacian have to do with random walks? How do you compute it?
So what would you need me to do to show that proving theorems is important? What would convince you? What is your criterion for a successful response?
Would we have GPS without relativity which in turn depends on non-Euclidean geometry, initially considered a curiosity? What about public key encryption, built upon number theory, proudly described as "useless" by the pure mathematician G.W. Hardy? There are so many examples like these.
Be careful of committing the fallacy in assuming that because a clear line of ideas can be traced back from the present day that finding that line was as easy as recounting it.
Out of interest, what is your position of the spending on sport, the arts, and other cultural endeavours?
My opinion on sports, arts etc. is the same. Publicly funded stuff is basically for the elite anyway, it's their way of cleverly siphoning tax-payer money (for there boring outdated interests which they partake in to show the illusion of sophistication). The masses pay to see their interests (and usually heavily taxed for it to boot).
So my point is this. I've read what you say, and I can see that you earnestly believe that pure mathematics and the theorems it produces are of no use. I'm asking what it would take to change your mind. What evidence would you need to convince you?
Science is grounded on falsifiable conjectures, and makes progress by testing those conjectures. If you are unwilling or unable to tell us what would be sufficient evidence, then I am unwilling to start chasing ghosts. It seems to me that trying to convince you would be like trying to nail fog to a wall.
Engineering is a critical discipline, and without it, things wouldn't get made. However, engineers rely on techniques that are known to work, and often that knowledge is based on deep theoretical work. Error-correcting codes, via which we get images back from Mars, and which allow reliable communications over cost-effective links are based on theoretical work. Yes, people played about and found the principles, but then they leveraged work done a century earlier to get to the limits. Having done so, they explored the theorems to see what axioms they were based on, and worked to see if those axioms could be circumvented.
It's the interplay between pure theory and pure experimentation that gives us enormous benefits, but it appears that you are willing, even eager, to dismiss fields of which you have no knowledge, purely because you can't see how they can possibly be useful.
It's Blub[0][1], all over again, but here we're not on a linear continuum, we're in a richly interconnected web of dependencies.
Another example. Fourier Transforms were first explored as a purely theoretical construct, showing that functions (with appropriate properties) form a vector space, and that vector space therefore has a basis, and that we should therefore be able to decompose functions into a representation relative to some basis. For decades this was a novelty, and then people started to use it for real. The theorems show the limitations, and then the engineers explore what can be physically achieved within those limits. Wavelets are now often used in contexts where the usual Fourier basis of trigonometric waves prove to be less useful, but the underlying theory is identical, and is still applied. And we know it will work, because of the theorems that were proven decades ago. The vector spaces are, by the way, infinite dimensional, and the work to understand the infinities was done as a part of pure math with no obvious applications.
Another example. We know that some problems are equivalent to others, and we know that the current best algorithms for these problems are exponential. We therefore know, for sure, and not just because of experiments, that some instances of some search spaces will be infeasible without a major break-through. As a specific example of that, multiple times people have claimed efficient algorithms for problems known to be NP complete. In some cases I've been able to prove that their algorithms, while possible useful in general, are definitely not polynomial. I can do that because of the theorems I have to hand.
But you won't care, and I can't make you care. I'm not writing this for you, because you appear not to be willing to change your mind, or consider that a field of which you appear to know very little might just be useful.
No, I'm writing this for people who read your comments and wonder. I'm writing this for people who are willing to entertain the idea that things of which they know little or nothing might be useful in ways they can't yet imagine.
[0] http://www.paulgraham.com/avg.html
[1] http://en.wikipedia.org/wiki/Paul_Graham_%28computer_program...
The example you gave of your proof doesn't say all that much, you won an argument (you said those algorithms were practically useful anyway).
I'm sincerely sorry if I come across as hostile. I'd just like to see mathematicians own up and change the corruption from the inside. A lot can come from it, you can change the world (you've taken up a lot it's resources, including some of the smartest people).
> You've given examples of old theorems, and also said
> they weren't actually useful at first.
> Later practical work lead to someone seeing use in them
> (but only the a vague way - limits - which could have
> been and probably found through experiment as well).
In my own work there are at least a dozen cases where knowing a theorem has provoked an exploration, that has then turned out to be useful. The investigation would most likely never have started, the techniques never suspected, without that initial theoretical knowledge. Again, engineering and pure math research in combination, but without the pure math to start with, some things might never be found. We can't know that, of course, but for those who have studied math, the connection is clear. For those who haven't, it's harder to see these things happen. It all seems so obvious in retrospect.
> The example you gave of your proof doesn't say all
> that much, you won an argument (you said those
> algorithms were practically useful anyway).
> I'm sincerely sorry if I come across as hostile.
> ... rarefied "breed" of pure mathematician earning a
> salary is as much of a waste as if billions were spent
> on buggy whip boys.
> ... it's the full employment act for pencil pushers.
> A lot in common with fundamental religionists.
> I never tried getting education in it in the first place.
> I had hunch it's useless and the older I get the more
> I think that was correct.
> It's justification for pencil pushers who don't want
> to do real work.
> I'd just like to see mathematicians own up and change
> the corruption from the inside.
> ... you've taken up a lot it's (sic) resources, including
> some of the smartest people.
Again, I'll never convince you, but I thought your comments should not stand without reply. Perhaps you are in earnest and genuinely believe the things you say, but you are arguing from a position of willful ignorance, which makes it hard to take you seriously.
[1] "Prediction is very difficult, especially about the future." -- Niels Bohr
Most theory I have read[2] on AI, machine learning, and knowledge deals in depth with the concept of symbols and symbol manipulation. Some would claim (correctly!) that computers are in essence simply symbol manipulation machines. The thing is, these symbols are really just ways of encapsulating a piece of information. Often, symbols are described as being 'made-up' of other symbols; symbols are used to describe other symbols. As a simple and obvious example:
- bits (low level 'axiomatic' symbols) are organised into groups to create bytes (still low level, not much meaning here yet)
- bytes are associated with letters (in an isomorphic relation which imparts meaning[3])
- letters are organised into groups to form words (a significant amount of meaning has now appeared)
- words are strung together in sentences
- sentences are woven into paragraphs/chapters/themes/poems/stories/epics
If I was to claim that a poem or story has no meaning simply because I can craft it from simple bits I would be missing something very important about symbol manipulation.
When I take a collection of ideas, snippets of information and learning, and am able to label them with a name I can transcend my current level of understanding and deal with ideas which are invisible on the lower level. For a programmer, this hierarchy of symbols presents itself as a hierarchy of low, medium and high level languages. Writing a python program in assembly would be absurd!(although perhaps appropriate in some cases)
Theorems (and Axioms) are the symbols of mathematics, and proving Theorems is symbolic manipulation in one of its purest forms. In fact, proving Theorems is the main way in which new 'meaningful'[4] symbols are created.
So why is proving Theorems useful? In short, because it is how new knowledge is created! (in this context, mathematical knowledge)
Is mathematical knowledge useful? I think anyone living in this modern world of ours (and I think that phrase applies to most people in most of humanity's history) can see that at least some mathematics is useful. Many have here argued about the different merits of developing mathematical knowledge (ie proving Theorems) before an application is well known, so I won't labour that point. Instead, I think I'll direct the interested reader to G.H. Hardy's 'A Mathematician's Apology'[5] for a great essay on the many merits a joy of mathematics can bring the suitably inclined soul.
If knowledge is a good thing, and proving Theorems increases knowledge, than proving Theorems is a good thing. To say it is better than other ways of spending our efforts is hard, but I think we can definitely agree that it is useful.
[1] My own definition!
[2] Which is tragically less than I would like
[3] Douglas Hofstadter in Gödel, Escher, Bach explains this idea so much better than I can - see https://en.wikipedia.org/wiki/Gödel,_Escher,_Bach
[4] An idea again borrowed from Hofstadter. We can string any string of words together that we want, doesn't mean that they mean anything to us. Formal system such as mathematics give us a way to combine symbols to give us new meaningful symbols. My claim that it is the main way they are created is founded on wild speculation and heavily biased historical observation.
[5] http://www.math.ualberta.ca/~mss/misc/A%20Mathematicians%20A...
Also during a Computational Complexity course, when we were studying de-randomization techniques, computing the Spectrum of an expander graph[2] and looking into the eigenvalues and eigenvectors of the graphs' adjacency matrix to understand topological properties from graphs, I realized how powerful and general Linear Algebra can be.
2d and 3d vectors over the Field of Real Numbers are easy to see, and a good way to get started. But many beautiful and powerful things come from this theory.
[1] http://en.wikipedia.org/wiki/Degree_of_a_field_extension
If anyone is interested in building an understanding of Linear Algebra from the ground up, I would suggest some of Israel Gelfand's work: http://www.scribd.com/doc/35985818/Gelfand-Lectures-on-Linea... It's not for the light hearted, but when you get through it, you will be deeply changed and have a new perspective on even basic math, such as integration.
If that's true, I'm sorry for the engineers who took this course.
I did not see a single number in my algebra course, neither 2d or 3d vectors. As a matter of fact, we followed the same program that math students (axioms, theorems, linear spaces, rings, ...). Even some teachers were teaching in both faculties, and our exams usually were more difficult (this was just a policy, there were too many engineering students).
Now, I'm in a different university where the maths future engineers study are much more "practical". It's sad seeing that students cannot understand, for example, that a principal direction in a stress tensor is just an eigen-value or that a rotation is just a change of base and, therefore, there are many theorems and properties you can apply to these concepts just making use of some basic algebra.
It's always good to know well the tools you have to use and, for an engineer, algebra is one of the most important tools.
Thanks for the link!
http://linear.ups.edu/xml/latest/fcla-xml-latest.xml#fcla-xm...
http://joshua.smcvt.edu/linearalgebra/book.pdf
Also, the common 1st year texts (Anton, Lang, Hoffman/Kunze and Friedberg/insel/Spence) can be found easily for cheap, used. The old edition of Strang I used to have was good too, but some people react really strongly when you bring it up. There's lots of ways ot sequence LA and the needs of EE's, econometrics/game theory, prob/stats and applied mathies are different from physics/math majors. Look at ToC's and read the Holyoke prof's writeup:
http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Gil...
(i think they're 1st year texts, my Dad's a physics prof and he started talking about determinants around 7th grade)
Other things that might be nice to include are dot products (projecting one vector onto another as a measure of co-linearity) and rotation matrices (you could keep it to 3x3).
I am looking for the kind of book that will make you fell in love with Linear Algebra. For Calculus, I used Piskunovs Differential and Integral Calculus which was miles away from what my classmates were using. That along with Maple help to double check that my stuff was correct proved a good combo. My current Linear Algebra book is an honest book but it is a boring book, it fails to entertain or to amaze or to give you those moments of insight that puts a grin in your face.
I think I just wish I had better books as an undergrad :-(
This is the first, simplest, off-hand example I can come up with.
Things are blocked "in parallel" (assuming the camera is far away from the objects) because that projection is a linear map (I repeat, assuming "far away focus") and has a one dimensional kernel: 2d + 1d = 3d, which means (flat TV + orthogonal lines = 3d world), that is the dimension formula.
http://nostarch.com/linearalgebra
:D
Do schools really not teach the underlying concepts of math, or do people just fail to understand them the first time through and then blame their teachers?
/took linear algebra in the math department, then TAed it.
It is absolutely the case that a great many students will not remember seeing material that they definitely were exposed to. As someone who's TA'd physics classes for a few years, I'd often ask a recitation whether they'd covered some particular subject yet in lecture, and they'd often say (as a class) no.
I'd ask the professor later, and of course they had -- but students rarely go into lecture having done the reading or prep work, and so have a piss-poor retention rate.
I've found that even if do the reading, my brain is often, well, a bit leaky... TT
Whether or not someone is sufficiently motivated to study something should not affect their ability to understand it. Vector spaces and linear maps are actually much easier to understand, and I think that at some point in mathematics (probably your first analysis course) your motivation has to come from the beauty of the theory itself rather than some real-world application.
Should they deserve to starve like artists then?
Should taxpayers fund disciples (students) for them?
If yes, than should taxpayers be forced to fund students for any cult anyone wishes to start?
You deserve a round of applause!
(Also, mods, where are you?)
What are you imagining when you hear "linear algebra"?
What will have more societal impact? Should anyone wishing to use a wordprocessor be required to understand turing machines and computer architecture?
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.
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.
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.)