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[flagged] No bullshit guide to linear algebra – v2 (minireference.com)
242 points by ivan_ah 39 days ago | hide | past | web | 72 comments | favorite



This book is neat, and I hate to sidetrack any thread on HN about linear algebra, but rules are rules, even for people I like.

You're not supposed to solicit upvotes for submissions. But that's exactly how this story on HN came to be: everyone who bought the book got an email announcement that the v2 version was finished (congrats, btw!) and it specifically asked people to upvote this particular submission.

Please don't do things like that. Just a few steps past that and you've got Product Hunt and the lesser subreddits. It's a good piece of work; people would have noticed and highlighted it on their own.


Mea culpa. I won't be doing that in the future (I mean writing books and posting them to HN once every two years, yes, but soliciting upvotes, no.)


Product Hunt and the Lesser Subreddits is my new favorite band.


this might be the most amazing book ever and I might even buy it and I know I'll get down votes because so many of you love books but....

part of me feels like , especially for a topic like this, I want webpages with interactive diagrams. Not last century's info dissemination tech but this century's info dissemination tech

I feel page like this

https://pomax.github.io/bezierinfo/

is far better online than any paper book on the same topic could ever be. that feels like it's probably true for almost all math, physics, and computer topics.

if I'm reading up on game ai i'd much rather have in page interactive examples than static text on a page.

this isn't in anyway meant as a comment on the book above itself. it's more a comment on tech/math/physics books in general

anyone else agree or am I alone in this way of thinking?


There exists a book already for Linear Algebra w/interactive diagrams (that explodes my CPU) http://immersivemath.com/ila/index.html

I just want accompanying lectures that build intuition like these https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... or even the author standing in front of a blackboard discussing each chapter in the book and posting the lectures on the book's page. Some author's have done this like https://cs.wheaton.edu/~tvandrun/dmfp/

If it must be 'immersive' I'd also rather have a self-contained interactive book I can keep locally with signed errata updates instead of having to rely on slow websites that may be abandoned by the author or shutdown for whatever copyright reasons or ninja edited by somebody with misinformation.

Edit: or a book with an online tutor, like the one MIT made for SICP http://icampustutor.csail.mit.edu/6.001-public/


'Wikipedia with intuitions' would beat out any single book trying to record them. My hope is that they one day expand their scope to not only cover knowledge but different ways of looking at it/understanding it.

Then for any given topic a given subject's 'lesson plan' could just follow a 'sequence' (a la LW) of excerpts reviewed for accuracy (in lieu of reading such excerpts out of a book) along with mediawiki/etc animations/images/videos that go along with those subjects.

I suppose it'd be a good idea to version those 'lesson plans' so later courses/syllabuses can benefit from newly-researched knowledge after it has been peer reviewed and added to a later 'version' of the curriculum (i.e. mitigating the 'quality control' issue of information shared online). And of course lecturing as it traditionally exists can still thrive albeit with a more freed focus for interactive learning (Q+A, practice problems in class, or whatever transpires to be the best tool to use in dedicated class time/lecture time)... It just seems like singleton books as knowledge containers are obsolete next to the potential of QAing existing 'open source' knowledge without the publisher/etc overhead.


> My hope is that they one day expand their scope [...]

The “they” who make Wikipedia are just volunteer contributors. Go make it happen!


I can't imagine pouring my time and energy into adding this type of content and having it be removed for $arbitary_but_qualifying_reason.


The "they" who decide the content classes which will be handled by the project (enforced as rules of the wiki) are nonprofit organizations I'll never be on the board of, so basically what cgriswald said.


Your original comment can be summarized as “I bet it would be great if a bunch of other people did a few tens of thousands of hours of volunteer work to make new tools for teaching math to me.”

Which as an idle speculative wish is fine as far as it goes, but has a vanishingly tiny chance of happening in whatever way you are envisioning. (Of course, various people around the world who are working on teaching math will continue to do their best, but without paying any attention to your specific desires.)

If instead you said “Wow, I have this great idea that would be worth a large team of people spending a few tens of thousands of hours on. Does anyone have an idea about what kind of prototyping, political organizing, or other work I can do to start on a project like that so I can make my dream a reality? Is there anyone I should talk to about this?” Etc. that would be a much more interesting basis for further discussion, and might actually lead somewhere.

The volunteers (and the tiny handful of paid staff) in charge of Wikipedia are certainly interested in new ideas for improving things. If you think it’s impossible that they’ll ever accept an improvement, and you aren’t interested in discussion, what’s the point in speculating about it?


Does anyone know of an online, public, shared mindmap type thing?


Funny you'd ask, I used to look into that a lot too. There wasn't one when I last checked, lemme know if you hear anything.


I'll second the recommendation on that video series: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

Highly, highly recommended for anyone who wants to actually understand linear algebra. If you're studying the material or need to use linear algebra in applications or are just curious watch those videos. It's not a huge time invest and it's a huge "force multiplier" to actually have an intuition for what's going on under the hood instead of seeing things as just collections of numbers and magical sequences of operations that "do stuff" for who knows what reason.


A proof-centric book I used that went well with the 3Blue1Brown series was Robert Valenza's LA book, which focuses on extending the applications of LA concepts to many settings which have no obvious geometric interpretations. It also has surprisingly easy proofs for exercises.


I was seriously confused why my Primer on Bezier Curves got hit for 500 visits in the course of an hour when it usually chugs along at a comfy 200~300 a day depending on which weekday it is... After finally going "fine, let's see if it's somewhere hidden in comments, let's hit up hn.algolia.com" turns out you're to blame... So thank you =)

I find that for a lot of Maths, having both a static textbook for "doing the rote exercises" and in paraller an interactive ebook or website is key for doing the thing you need to do to learn it, seeing the thing to feel it, and then ultimately step three being doing it yourself and seeing it work in order to understand it.

My page actually started that way - I wanted to understand Bezier curves because I was doing OpenType Font work from the ground up, and didn't know enough about Beziers, so I grabbed me some internet, learnt me some Bezier maths, and then in order to get the feeling and understanding working, I programmed the visuals so I could see what they did and in the process understand exactly why they did that.

Having pages like the Primer available is invaluable, but I'd posit that unless you also extend yourself to reading the "dry bits" and try to make it work yourself, it's really more eye candy than educational material. Nothing wrong with eye candy of course, and maybe it'll spark some recreational maths, but if you're serious about learning a subject, hit up all the resources and then try to teach it, even if just to yourself =)


Totally, one-hundred-thousand percent agreement from me.

Bret Victor liked to point out that we're completely ignoring the potential of the medium we're dealing with. We're using a dynamic medium made of magic ink to present static, non-interactive content. We're just delivering "information to be consumed" instead of creating "an environment to think in"[0].

If picture is worth a thousand words, then an interactive model is worth a thousand pictures. This is one good thing that could come out of proliferation of features in our browsers, as well as make tablets actually useful as a learning tool - that is, if more people bothered to use the medium to its full power. There are only few of those who try; some of their works can be found in [1], which is a lovely resource to sink your time in and see the yet untapped potential of digital technology.

--

[0] - http://worrydream.com/ExplorableExplanations/

[1] - http://explorableexplanations.com/


As hackermailman already pointed out, the book by Ström, Åström, and Akenine-Möller has some amazing LA visualizations: http://immersivemath.com/ila/index.html

Highly recommended!


`oncontextmenu="return false;"` on the body element. Sigh...


I love learning math from books, I feel like being forced to do the visualization in my own head can be helpful. I've seen a ton of beautiful math visualizations and I always enjoy them but on the whole, I think I've learned more from textbooks.

That said, the three blue one brown youtube course on Linear Algebra is truly amazing. I highly recommend it.


I agree completely.


The phrases "no bullshit" and "linear algebra" in the same sentence immediately brought to mind the essay "Down with Determinants!" by Sheldon Axler[0]. Having skimmed through the extended preview it looks like your focus is on the bare techniques and their applications rather than a playful approach, and determinants are introduced somewhat out of the blue (just as they are in almost any degree course). I realise of course that that's just a preview and the real thing could be very different. But I wonder what you think of the idea of postponing determinants until well after eigenvalues, diagonalisation and so on, when their definition and utility might be more appreciated?

[0] http://www.axler.net/DwD.html


Disclaimer: I am very happy that books like this are written, and look forward to never seeing another $120 doorstop being used as a text for the course anywhere. What follows is criticism and a question which, if addressed, could help those on the fence about this book.

----------------------

Speaking of Axler, his Linear Algebra Done Right is still the best book on the subject, in my opinion.

For one, it's lean. You can fit the softcover in a purse, and read it in the bathroom. It may be scant on examples, but it's a book that can be read in its entirety.

This book, at 500+ hundred pages, simply means nobody will really read it. No student is going to be reading several dozen pages of hard math EVERY WEEK - the exceptions to that are those who will do well with any other book too.

So we have a book here that, alas, is not meant to be read - in entirety, at least.

Yes, these are two very different books. But then we also have Linear Algebra Done Wrong.

A text by Serge Treil, it was written as a counter to Axler's book (and was named in jest). It covers many of the same topics, and is available as a free (!) PDF.

So, my question is - why this book? Aside from the title, what does it really do in a new way?

There's not enough in the preview for me to conclude that this book did anything Linear Algebra Done Wrong does not, so I ask those who read it (or the author).


Thanks for your question, which I will respond to in a second. First let me explain how we got to 550 pages. I wanted to include a review of prerequisites material from high school math, which burned 100pp right off the batt. Then the main material (Ch2-Ch6) is just 220pp which is not bad (especially when you take into account the book is a compact format 5.5" x 8.5"). All of this was done in 2014, when I started to think about the applications of linear algebra.

But then the feature creep starts! In the introduction of the book I make the claim about how powerful LA techniques are, and how they serve as the foundation of machine learning, and generally "scientific modelling." To substantiate this claim, I decided to make a kick-ass applications chapter that SHOWS how powerful linear algebra really is (show me, don't tell me). Two years later, the result is a 270-page survey of pretty much everything I've learned over the years from engineering, physics, and computer science that used linear algebra.

> why this book? what does it really do in a new way?

The main point of this book is to teach all the LA material in an approachable, informal narrative as opposed to formal the theorem-proof approach used in most textbooks. Can serious mathematical topics be introduced using informal language, yet without saying anything outright wrong? Can I trick the reader into learning advanced topics like Fourier transforms and quantum mechanics by making the connections to basic operations explicit? Can I sneakily introduce spaced repetition by covering core ideas several times in the book using different notation and in different contexts?


Thanks for the answer!

So, the book is really three books.

How do you get around actually proving things?


The book proves most things, just doesn't follow the standard math format of theorem-followed-by-proof. All the definitions are reasonably clear, and most of the proofs that are accessible and educational are given (e.g. the equivalence statements about invertible matrices). The exercises and problems also ask readers to complete several other interesting proofs (with solutions in the back).

In a future pass I plan to add more proofs, wherever adding formal structure doesn't interfere with the flow.


I've been trying to develop my maths skills beyond the high school level I was last formally educated at. My problem is, that was 20 years ago, and I've forgotten a lot of the fundamentals, but every time I try to slog through Khan Academy I get bogged down in what would be many, many months of review to get to the undergraduate level. Can anyone recommend materials that contain a high level overview of those rules and concepts that are taught at the pre-calculus level, so I can jog my memory without endless manual calculation? Linear algebra and statistics are mainly what I'm interested in learning.

Edit: This book looks like a good candidate. :)


Start with Strang's MIT OCW Linear Algebra course: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...

You really only need to know arithmetic to get started.

Use Anki to create flash-cards so that you learn faster and don't forget.

https://apps.ankiweb.net/

Also, manual calculation is fundamentally how you learn mathematics. Following along is a first step, but is not good enough to learn the material. You need to do the calculations and run into problems, so you can re-create the knowledge in your brain.

EDIT> Take Coursera's Learning How to Learn Course. It sounds flaky, but will accelerate all of your other learning, and help with procrastination as well.

https://www.coursera.org/learn/learning-how-to-learn


If you not familiar with Professor Leonard check out his playlists. I like his teaching style more than Khan.

https://www.youtube.com/channel/UCoHhuummRZaIVX7bD4t2czg


I'm in the same situation (I literally had to start with "what is a factor again?") and I've found just doing a little every morning works. It's slow but learning one or two concepts a day avoids frustration. Selby's Practical Algebra was good and then worked my way through Schaum's Precalculus book, supplemented by stuff on the web.


Fror a high-level overview of linear algebra, 3Blue1Brown's youtube series I cannot recommend highly enough.

https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQ...


For stats, I found the Stats for Dummies and Statistics in a Nutshell were great resources to refresh and review.

It's easier to relearn some of this stuff when you know what you don't know!


Lang's _basic mathematics_


Anyone who recommends Lang as a learning resource should burn in their version of hell.


Really? I've used it before for others who found it useful. That's the thing about math, lots of options/styles. Just look at Spivak vs Apostol for Calc. Do you have a suggestion for him?


Lang's Algebra book before revisions contained famously difficult exercises https://mathoverflow.net/a/10899 but I liked Basic Mathematics even if it's out of print (expensive) and full of errata issues.


This is the exact sort of book I've been looking for for a long time. I work with quite a few physicists and engineers who make extensive use of linear algebra I don't fully understand (my university courses only went up to chapter 6.1's content), so I've been looking for some way to get up to speed.


This is exactly like every other linear algebra book I've ever seen, other than the title. I was hoping this had been improved over v1 but it seems to be unchanged in any significant way. What exactly does the author think the bullshit is that they're leaving out?


I bought the first edition of this book and I enjoyed a lot.

Ivan, can you give those of us with the first edition a summary of the changes?


The "core material" is pretty much the same. It covers all the computational, geometric, and theoretical aspects of linear algebra as required for a first-year course.

The main difference is the applications. Instead of one chapter, I expanded this part to three chapters that cover applications to chemistry, electronics, economics, optimization, probability theory, and even quantum mechanics. All of these sounds like fancy stuff, right? Actually they're simple applications of basic ideas of vectors and matrix operations, so it's quite easy to pick up once you learn LA.

BTW, if you bought the book the pre-release version through gumroad, you have access to v2 as well. Check your email—I just sent out an update.


You might want to sell a standalone "basic high school math" book and then devote more pages to linear algebra etc. I bought No BS Calc and Physics and while I liked the fact that both were taught together, I felt like some concepts were underexplained.


Do you plan to add one on graphics?


There is already a section on computer graphics[1] that covers basic homogenous coordinates (instead of representing points as 3-vectors [x,y,z] you represent them as 4-vectors [x,y,z,1], which allows you do do translations and projections easily).

I explain the math, but I'm not expert enough to talk about practical graphics topics like OpenGL / WebGL etc, so not sure how much further I can go. To merit the "NO BULLSHIT" label, it's really important that the author know the subject matter inside out ;)

[1] https://minireference.com/static/excerpts/noBSguide2LA_previ...


Thanks! That looks good. I didn't mean any extensive chapter on OpenGL and the like, just some examples of linear algebra commonly used in computer graphics.


As if other books on linear algebra have bullshit... This contributes to the "math sucks" culture. Math is fun an easy to learn. It just takes effort, like any other skill.


You may disagree but most books I read on math, or subjects that require math, contain a sizable amount of BS. Not in the sense of containing things that are wrong but in the sense of containing things that are there for the author to show off rather than for the reader to comprehend the subject matter.

I read books to learn about something that I didn't know about before. So I as a reader can't distinguish between the important stuff and the 'fluff'. The author has to do that for me. The best texts do that rigorously but the bulk does not, unfortunately.


> Math is fun an easy to learn. It just takes effort, like any other skill.

For you, and for me, but not for everyone.

Why worry about the title if it brings more people to the table?


The essence of linear algebra series on youtube is great as well. It provides a intuitive guide to what linear algebra is about in 2d and 3d space with "arrow" vectors with great animations. Once an intuition is provided about matrix transformations, determinants, basis changes and eigenvectors, then it explains how abstraction of this theory can allow carry over of these insights and techniques to other domains or other abstract vector spaces.

Too often learning starts at the abstracted level so intuition is lost in exchange for memorized techniques. There is a flip side of intuition-first techniques basically dumbing down everything into tweets but this video series is great at not doing that IMO.


I own a physical copy of Savov's "No BS Guide to Math and Physics" and am very pleased with it. I found it particularly useful as a refresher on things I learned long ago, but no longer regularly use. The focus on practical application is of particular value in a quick reference role. I've also recommended it as a reference for students transitioning from a Technologist program to formal studies in Engineering. I've ordered this latest work just on the basis of that experience (and the recent specter of not remembering how to use eigenvectors when it come up in an obscure graphics problem).


Just bought this. This book looks awesome — I'm working on some image processing this summer and I want to re-strengthen my Linear Algebra. It seems to be foundational for everything going on these days.


Congrats Ivan, this was long in the making!


Yeah, it seemed like it would never end. Thx for letting me use the vertical aspect ratio monitor... that helped a lot!


Ivan's books are always in my reference library. He is the Silvanus P Thompson of our modern day. Looking forward to reading Linear Algerbra and as always learning something new. Jjcona


"concise, conversational tone" I wonder which it is. :-p


"Check this shit out:" in Section 1.1, page 9 of https://minireference.com/static/excerpts/noBSguide2LA_previ...


It's actually pretty nice. I was mostly just being silly. But I do think concision is underrated, and some "conversational" tutorials drag on… Mathematicians end up with terse notation on purpose.


I took a course in Computational Biology last semester and we covered all of a typical one semester linear algebra course in half the semester. It moved so fast that no one was really able to comprehend the material well enough, so I never felt like I actually learned the material as much as I was exposed to it and just understood its importance. I will definitely check this out as it may possibly be what I need to finally understand linear algebra. Thanks for all the hard work!


The sample doesn't really give me a sense of whether this book is right for me. It contains a lot of material from the early chapters, but little of the parts of that confused me in college.

My primary question is whether this book is good at conveying the reasoning that promotes intuition around the more difficult (for me) topics, such as vector spaces and the many theorems associated with the Fundamental Theorem of LA?

If not, suggestions for other resources are very welcome!


This looks fantastic! I'm a high school grad with a 2 year college degree but in college I took only a single math course (I was required to repeat Algebra 2 because my college didn't accept my high school's transcript for whatever reason) and I've felt behind from many of my peers not knowing enough "maths".

I need to brush up on my former knowledge and this look slike a great way to learn more!


Ivan. in the refresher sections, after each section, for example, Complex Numbers, there is a list of sections in the example PDF, is this expanded in the real book or are they topics for further reading? I notice similar sections in earlier subjects seem to be expanded.


Yes, all sections in the real book are expanded. The idea for the preview was to leave only the headings so that the table of contents will look the same.


Cool, picking up a copy now. I love the way it's written but unfortunately not safe to read to my kids as a bedtime story.


Wow, on demand shipping is fast. I already have it.


Just bought two copies, one for me and one for my son.

Nice textbook in Computer Modern just gets me all jazzed up. :)


Great book that will fill a great void in a topic so fundamental for moving on to more advanced subjects.


Please present one example of a linear algebra text which does contain "bullshit".


You're right. Of all the mainstream textbooks on subjects for first-year science, linear algebra textbooks are the best of them all.

I guess the main "bullshit" I could reproach is the price tags, $60 -- $90, but even that is nothing compared to the textbooks for first-year calculus and mechanics that run $200+.


The extended previews and concept maps are a great idea.


Good job Ivan, planning to purchase!


Well done Ivan - I think I might purchase this :-)


(1) Can work with any algebraic field. The greatest interest in practice is for the field the set of real numbers R. The set of complex numbers C can also be important. Curiously, there are some applications, say, error correcting codes, where the field is finite, e.g., the set of integers modulo a prime number.

For learning, I recommend working with the complex numbers C for where they do help, with very little extra effort, and let the real case be mostly a special case.

(2) The main motivation for linear algebra is just solving systems of simultaneous linear equations as in high school. The main solution technique is just the elementary row operations used as in Gauss elimination -- can learn that in three minutes and program it in 10.

(3) Matrix theory is central, even crucial, in linear algebra but really is just some darned convenient notation that can motivate just from solving a system of linear equations. So, for positive integers m and n, an m x n matrix is a rectangular array of numbers (real, complex, other, as you have decided to work with) with m rows and n columns. So, it's just notation.

(4) The big deal about matrix theory is just that can multiply matrices. The multiplication is defined just so that a system of linear equations can be written as a product of (A) a matrix from the coefficients and (B)+ a matrix with one column of the unknowns set equal to a matrix of one column for the right hand side. Let's try it for some high school case of 2 equations in two unknowns (roughly as in TeX, and underscore character starts a subscript):

     a_11 x_1 _ a_12 x_2 = b_1

     a_21 x_1 _ a_22 x_2 = b_2
So, the 2 unknowns are x_1 and x_2.

                       / a_11  a_12 \
     2 x 2 A =  |                        |
                       \ a_21  a_22 /

                      / x_1 \
     2 x 1 x = |  x_2 |
                      \ x_3 /
and similarly for 1 x 1 b. Then the matrix multiplication Ax is defined just so that we can write Ax = b and have that be equivalent to the 2 equations in 2 unknowns in the high school form above.

Commonly we write A = [a_ij] for i = 1, 2, ..., m and j = 1, 2, ..., n.

Now, we can do some arithmetic on the matrices, and that starts to get darned useful: E.g., for a number p we can define the product pA = [p a_ij]. So we multiply the matix A by the number (call it a scalar if you want) p. And we can add two matrices: So for m x n A = [a_ij] and m x b B = [b_ij] we can define m x n C = A + B where C = [a_ij + b_ij]. So just add the corresponding components (elements, entries).

Then presto, bingo, we get to why this subject is called linear algebra: Given m x n A, n x 1 x, n x 1 y, and numbers p and q, we have

     A(px + qy) = (pAx) + (qAy)
which says that A is linear or a linear operator -- in calculus, both differentiation and integration are also linear operators. There is a huge book Linear Operators by Dunford and Schwartz, that is, functional analysis, that is where the points we are talking about, analyzing, actually functions. So, biggie special cases are Hilbert space and Banach space. So, linearity in linear algebra is a biggie start on functional analysis.

Why care about linearity? Because its (A) simple, (B) permits powerful theorems fairly easy to prove, and (C) is an assumption that often holds quite well in applications in science and engineering.

The next biggie deal about matrix multiplication is that it's associative:

     (AB)C = A(BC)
and that property drives a lot in duality, game theory, matrix computations, error correcting coding theory, etc. Can write out a proof in terms of the i's and j's and, there, will see that the crucial step is being able to interchange the order of summation signs (which in calculus is the same as interchanging the order of integral signs which in more advanced parts of calculus is Fubini's theorem) when adding finitely many numbers but could also know that the result is true because A, B, and C are functions, function composition is essentially just matrix multiplication, and function composition is essentially obviously associative!

Given m x n matrix A with m = n, then A is square. If a_ij = a_ji, that is, if reflect about the diagonal from upper left to lower right and get the same thing, the A is symmetric.

Now a biggie, IMHO the most important result in linear algebra: Regard


Mod: Please delete this post and its second copy. Both were posted while in draft and in error. I had to stop working on the post and just left the Web page without ever clicking on Submit, and somehow two copies got posted.

Maybe in the future I will complete the writing and post, but what is there now is incomplete and too rough. So, please delete.


(1) Can work with any algebraic field. The greatest interest in practice is for the field the set of real numbers R. The set of complex numbers C can also be important. Curiously, there are some applications, say, error correcting codes, where the field is finite, e.g., the set of integers modulo a prime number.

For learning, I recommend working with the complex numbers C for where they do help, with very little extra effort, and let the real case be mostly a special case.

(2) The main motivation for linear algebra is just solving systems of simultaneous linear equations as in high school. The main solution technique is just the elementary row operations used as in Gauss elimination -- can learn that in three minutes and program it in 10.

(3) Matrix theory is central, even crucial, in linear algebra but really is just some darned convenient notation that can motivate just from solving a system of linear equations. So, for positive integers m and n, an m x n matrix is a rectangular array of numbers (real, complex, other, as you have decided to work with) with m rows and n columns. So, it's just notation.

(4) The big deal about matrix theory is just that can multiply matrices. The multiplication is defined just so that a system of linear equations can be written as a product of (A) a matrix from the coefficients and (B)+ a matrix with one column of the unknowns set equal to a matrix of one column for the right hand side. Let's try it for some high school case of 2 equations in two unknowns (roughly as in TeX, and underscore character starts a subscript):

     a_11 x_1 _ a_12 x_2 = b_1

     a_21 x_1 _ a_22 x_2 = b_2
So, the 2 unknowns are x_1 and x_2.

                       / a_11  a_12 \
     2 x 2 A =  |                        |
                       \ a_21  a_22 /

                      / x_1 \
     2 x 1 x = |  x_2 |
                      \ x_3 /
and similarly for 1 x 1 b. Then the matrix multiplication Ax is defined just so that we can write Ax = b and have that be equivalent to the 2 equations in 2 unknowns in the high school form above.

Commonly we write A = [a_ij] for i = 1, 2, ..., m and j = 1, 2, ..., n.

Now, we can do some arithmetic on the matrices, and that starts to get darned useful: E.g., for a number p we can define the product pA = [p a_ij]. So we multiply the matix A by the number (call it a scalar if you want) p. And we can add two matrices: So for m x n A = [a_ij] and m x b B = [b_ij] we can define m x n C = A + B where C = [a_ij + b_ij]. So just add the corresponding components (elements, entries).

Then presto, bingo, we get to why this subject is called linear algebra: Given m x n A, n x 1 x, n x 1 y, and numbers p and q, we have

     A(px + qy) = (pAx) + (qAy)
which says that A is linear or a linear operator -- in calculus, both differentiation and integration are also linear operators. There is a huge book Linear Operators by Dunford and Schwartz, that is, functional analysis, that is where the points we are talking about, analyzing, actually functions. So, biggie special cases are Hilbert space and Banach space. So, linearity in linear algebra is a biggie start on functional analysis.

Why care about linearity? Because its (A) simple, (B) permits powerful theorems fairly easy to prove, and (C) is an assumption that often holds quite well in applications in science and engineering.

The next biggie deal about matrix multiplication is that it's associative:




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