[HN Gopher] Linear Algebra Done Right - 4th Edition
___________________________________________________________________
Linear Algebra Done Right - 4th Edition
Author : __rito__
Score : 244 points
Date : 2023-10-29 16:31 UTC (6 hours ago)
(HTM) web link (linear.axler.net)
(TXT) w3m dump (linear.axler.net)
| threatofrain wrote:
| Note that Axler intended this book to be the _second_ reading of
| Linear Algebra after you 've already taken a first course, but it
| is doable for a first reading.
|
| If you want to be crazy you can also check out A (Terse)
| Introduction to Linear Algebra by Katznelson & Katznelson.
| uoaei wrote:
| Dang, I remember everyone not enjoying linear algebra class
| with Katznelson in undergrad. I did ok but it felt like way
| more focus on things like row elimination algorithms than _why_
| any of it works. It wasn 't until I worked with a PhD geometer
| that any of it made sense and they largely cribbed from _Linear
| Algebra Done Right_. Hopefully the book is better than a class
| aimed at a generic mix of STEM undergrads.
| tptacek wrote:
| I did undergrad linear algebra with my daughter last semester,
| and Strang and Axler were a good one-two punch, Strang for the
| computation, Axler for the proofs homework.
| basedbertram wrote:
| So did you read through Strang and then read Axler, or did
| you try to work through both of them at the same time?
| tptacek wrote:
| Same time. I'd taught myself linear algebra from the Strang
| lectures (and a Slack study group we set up with some
| random university's syllabus, which gave us a set of
| homework problems to do) long before this, so mostly I just
| matched the professor's lectures to the Strang material,
| and dipped in and out of Axler when proof and conceptual
| stuff came up; it's not like we did Axler cover to cover.
|
| Before doing this, I'd only ever sort of skimmed Axler;
| it's sort of not the linear algebra you care about for
| cryptography, and up until the spectral theorem stuff
| that's exactly what Strang was. It was neat to get an
| appreciation for Axler this was.
| qbit42 wrote:
| Yeah, my math class followed Axler, which was great - but I
| didn't really get a feel for how useful linear algebra was
| until I read through Strang on my own. The applications are
| endless!
| tripdout wrote:
| Wow, we used this textbook (albeit alongside a more beginner
| focused one) in first year undergrad Computer Science.
| febra_ wrote:
| I have this book. It's been more than great. I fully recommend it
| to anyone looking for a quick and simple introduction to linear
| algebra
| some_math_guy wrote:
| Like basically everybody else I teach out of this book, and I'm
| happy to see a new edition. I'm curious what's changed/added -- I
| already am unable to get through the whole thing in a semester.
|
| At our school students take a computational linear algebra course
| first (with a lot of row reduction). So I am slowed down a bit by
| constantly trying to help the students see that the material is
| really the same thing both times through. I do wish there were a
| little more of that in Axler.
| blovescoffee wrote:
| I've studied from this book. Since you're teaching out of it,
| I'm curious if you've read/have an opinion on Strang's books. I
| love his lectures :)
| some_math_guy wrote:
| Sure, I am very familiar with them -- I actually TAed 18.06
| for Strang once upon a time. They're great books too. Which
| is better is mostly a question of what point of view you're
| after -- if you want to actually calculate anything, Axler's
| book is not going to help you, but if you want a more
| conceptual view of the subject it's best place. If you're
| really serious about learning linear algebra, you probably
| want to read both, first Strang, then Axler.
| seo-speedwagon wrote:
| The 3rd edition of this book is what my undergraduate linear
| algebra course used. It was a fantastic book. I feel like more
| computation- and determinant-heavy approaches can make the
| subject feel like a slog, but this book made me really enjoy,
| appreciate, and get a gut-level understanding of the subject
| gmiller123456 wrote:
| It's not at all obvious from the headline, but the news is that
| the book is _FREE_ , you can download the PDF from the first
| link.
| cyberax wrote:
| I skimmed it, and it's a bit too succinct for my liking. I guess
| it's inevitable if you want to try to pack that much material
| into one book.
| nextos wrote:
| I dislike the typesetting changes that came after the second
| edition.
|
| It was a really elegant book, reminiscent of other classic
| Springer Undergraduate Mathematics Series tomes.
|
| Lots of distracting color, highlighting, and boxes were added,
| which IMHO make the book less clear.
|
| Obviously, the content is still great.
| fiforpg wrote:
| Yep. Huge kudos to the author for making it available, but the
| PDF does feel sloppy with all the bright colors and images. In
| science textbooks, less is more.
|
| Compare this to the very latest edition of Stewart's calculus,
| which now uses even more pastel, subdued colors for diagrams.
| nextos wrote:
| Exactly, a calm black and white design does not need to be
| unfriendly.
|
| Hubbard & Hubbard or MacKay are two examples of beginner-
| friendly books with great typesetting.
| blt wrote:
| with the power of `tcolorbox` comes great responsibility.
| __rito__ wrote:
| This book is Open Access and you can download it from this link
| [0].
|
| [0]:
| https://link.springer.com/content/pdf/10.1007/978-3-031-4102...
| sieste wrote:
| The look on Christina of Sweden's face on page 1 made me laugh,
| she looks exactly how you would imagine a 17th century princess
| hearing about Linear Algebra.
| navanchauhan wrote:
| For the course "Linear Algebra for Math Major", we are using both
| LADR and Linear Algebra Done Wrong (Open Access as well)
| readthenotes1 wrote:
| Linear Algebra Done Right This Time We Promise (4th try)
|
| J/k ;)
| mike986 wrote:
| I have surveyed every LA books out there and a lot of amazons
| reviews claimed axler's book is the best LA book.
|
| It might be for case for printed books for sale. But I stumbled
| upon Terrance Tao's pdf LA lecture slides on his website and it
| is so much better than all the books I've surveyed.
|
| The writing is super clear and everything is built from the first
| principles.
|
| (BTW terry's real analysis book did the same for me. Much more
| clear and easy to follow than the classics out there)
| abdullahkhalids wrote:
| I believe these are notes that you are referring to
|
| https://terrytao.files.wordpress.com/2016/12/linear-algebra-...
| pstuart wrote:
| Thanks!
| zeroonetwothree wrote:
| I'm not sure that Axler's book is great as a _first_ LA book. I
| would go with something more traditional like Strang.
|
| Although I really didn't feel like I "got" LA until I learned
| algebra (via Artin). By itself LA feels very "cookbook-y", like
| just a random set of unrelated things. Whereas in the context
| of algebra it really makes a lot more sense.
| mohamez wrote:
| >I'm not sure that Axler's book is great as a first LA book.
|
| Linear Algebra Done Right is a text for beginners who want to
| study linear algebra in a proof based, mathematically
| rigorous way.
|
| So, if you want that I think it's a good fit as a first
| linear algebra book.
| ayhanfuat wrote:
| From "Preface to Students":
|
| > You are probably about to begin your second exposure to
| linear algebra. Unlike your first brush with the subject,
| which probably emphasized Euclidean spaces and matrices,
| this encounter will focus on abstract vector spaces and
| linear maps. These terms will be defined later, so don't
| worry if you do not know what they mean. This book starts
| from the beginning of the subject, assuming no knowledge of
| linear algebra. The key point is that you are about to
| immerse yourself in serious mathematics, with an emphasis
| on attaining a deep understanding of the definitions,
| theorems, and proofs.
|
| It is definitely a hard text if you haven't had exposure to
| linear algebra before.
| SamReidHughes wrote:
| The thing is, by the time you get to this book, most
| students have probably taken DiffEq or multivariable
| calculus, and had exposure to linear algebra there. (If
| not in high school.)
| agumonkey wrote:
| talking about amazon, someone suggested me to get gareth
| williams linear algebra with applications (5 bucks on ebay)
|
| it's a good applied primer, not big on concepts, more about the
| mechanics, and it unlocked a lot of things in my head because
| dry textbook morphisms definitions sent me against imaginary
| walls faster than c
| JadeNB wrote:
| Because the poor guy contributes so much to math and math
| exposition and yet has his name misspelled everywhere, I'll
| mention that it's Terence, not Terrance.
| SAI_Peregrinus wrote:
| Have you seen Macdonald's "Linear and Geometric Algebra"? I
| found it a much nicer introduction to the subject.
| fiforpg wrote:
| About avoiding determinants to the degree that this book does:
| while I agree it makes sense to delay introducing them, the goal
| should not be avoidance but clarity. The way author has to bend
| himself backwards here when dealing with eigenvalues isn't great
| either.
|
| I would recommend Strang for a healthy balance in handling
| determinants.
| jjoonathan wrote:
| I read Strang and then Axler. Strang is great at numerics but
| weak at presenting the abstract picture. I feel like if I had
| taken, say, finite elements (or any other subject where it's
| important to take the abstract / infinite dimensional picture
| seriously before reducing to finite dimensions) right after
| Strang without reading LADR then I'd have been seriously
| underprepared.
| fiforpg wrote:
| You have a point in that to understand any particular subject
| well, it makes sense to read _more than one_ book on it, at
| least to compare the different perspectives.
|
| Also worth noting that Strang has a couple of similar linear
| algebra books, so we might not even be discussing the same
| text.
| jjoonathan wrote:
| That's entirely possible, but in the context of
| introductory books I think it's fair to assume & limit
| scope to Strang's "Introduction to Linear Algebra" and
| Axler's "Linear Algebra Done Right."
|
| I am an applications-oriented person and my inclination was
| to go directly from a matrix/determinant heavy picture into
| applications. Strang['s intro text] only. I am extremely
| glad that someone intercepted me and made me get some
| practice with abstract vector spaces, operators, and inner
| product spaces first, using Axler. This practice bailed me
| out and differentiated me from peers on a number of
| occasions, so I want to pass down the recommendation.
| CamperBob2 wrote:
| Honestly, I think Strang is overrated. Yeah, I know, on HN
| that's like criticizing Lisp or advocating homebrew
| cryptography or disagreeing that trains fix everything. But
| still.
|
| I bought his 6th ed. Introduction to Linear Algebra textbook,
| and he doesn't get more than two pages into the preface before
| digressing into an unjustified ramble about something called
| "column spaces" that appears in no other reference I've seen.
| (And no, boldfacing every second phrase in a math book just
| clutters the text, it doesn't justify or explain anything.)
| Leafing through the first few chapters, it doesn't seem to get
| any better.
|
| The lecture notes by Terence Tao that someone else mentioned
| look excellent, in comparison.
| ayhanfuat wrote:
| His lectures are great but I definitely agree about the book.
| It reads like one of the TAs transcribed the lectures and
| added some exercises to the end.
| noqc wrote:
| In my experience, it's a little bit easier for new students
| to understand that the image of a matrix is the span of its
| columns, hence column space.
| CamperBob2 wrote:
| Perhaps, but that's about as useful as pointing out that
| monads are a monoid in the category of endofunctors. What's
| the "image of a matrix?" Coming at LA from a 3D graphics
| background, I've never heard that term before. And what
| does the "span of its columns" mean?
|
| To me, each column represents a different dimension of the
| basis vector space, so the notion that X, Y, and Z might
| form independent "column spaces" of their own is
| unintuitive at best.
|
| These are all questions that can be Googled, of course, but
| in the context of a coherent, progressive pedagogical
| approach, they shouldn't need to be asked. And they
| certainly don't belong in the first chapter of any
| introductory linear algebra text, much less the preface.
| noqc wrote:
| Axler is pathological in his avoidance of determinants. I've
| heard (third hand) that he once pulled aside some fields
| medalist into a classroom after a talk and asked them "Do you
| like determinants?" I imagine him drawing the curtains and
| sweeping for bugs first.
|
| I attended a (remote) seminar where he was talking about this
| book, and this seems more or less accurate. Mathematicians are
| a weird lot.
|
| The response that he received in the story was "I feel about
| them the same way I feel about tomatoes. I like to eat them,
| but other than that, no, I don't like them."
| imjonse wrote:
| From the preface. "You cannot read mathematics the way you read a
| novel. If you zip through a page in less than an hour, you are
| probably going too fast." Sadly, he's probably right.
| zeroonetwothree wrote:
| Meh, there's different goals you could have. I actually find it
| enjoyable to read math more quickly (almost like a novel) which
| gives you a good sense of a lot of the higher-level themes and
| ideas. Then if it's interesting I might spend more time on it.
| mohamez wrote:
| He is talking about reading it after you decided that it is
| interesting.
| mohamez wrote:
| Tips on Reading Mathematics[1]:
|
| - Be an active reader. Open to the page you need to read, get
| out some paper and a pencil.
|
| - If notation is defined, make sure you know what it means.
| Your pencil and paper should come in handy here.
|
| - Look up the definitions of all words that you do not
| understand.
|
| - Read the statement of the theorem, corollary, lemma, or
| example. Can you work through the details of the proof by
| yourself? Try. Even if it feels like you are making no
| progress, you are gaining a better understanding of what you
| need to do.
|
| - Once you truly understand the statement of what is to be
| proven, you may still have trouble reading the proof--even
| someone's well-written, clear, concise proof. Try to get the
| overall idea of what the author is doing, and then try (again)
| to prove it yourself.
|
| - If a theorem is quoted in a proof and you don't know what it
| is, look it up. Check that the hypotheses apply, and that the
| conclusion is what the author claims it is.
|
| - Don't expect to go quickly. You need to get the overall idea
| as well as the details. This takes time.
|
| - If you are reading a fairly long proof, try doing it in bits.
|
| - If you can't figure out what the author is doing, try to (if
| appropriate) choose a more specific case and work through the
| argument for that specific case.
|
| - Draw a picture, if appropriate.
|
| - If you really can't get it, do what comes naturally--put the
| book down and come back to it later.
|
| - You might want to take this time to read similar proofs or
| some examples.
|
| - After reading a theorem, see if you can restate it. Make sure
| you know what the theorem says, what it applies to, and what it
| does not apply to.
|
| - After you read the proof, try to outline the technique and
| main idea the author used. Try to explain it to a willing
| listener. If you can't do this without looking back at the
| proof, you probably didn't fully understand the proof. Read it
| again.
|
| - Can you prove anything else using a similar proof? Does the
| proof remind you of something else? -
|
| - What are the limits of this proof? This theorem?
|
| - If your teacher is following a book, read over the proofs
| before you go to class. You'll be glad you did.
|
| [1] Reading, Writing, and Proving: A Closer Look at Mathematics
| By Ulrich Daepp and Pamela Gorkin.
| reader5000 wrote:
| I think in the modern era a very good piece of advice,
| particularly for those of us without gorilla-like stamina to
| comb through a math text, is to go on your favorite video
| website and watch through multiple videos on the topic.
| photochemsyn wrote:
| This is linear algebra for undergraduate math majors, but if you
| just want an basic understanding of the topic with a focus on
| computational applications, Poole's "Linear Algebra: A Modern
| Introduction" is probably more suitable as it's heavy on
| applications, such as Markov chains, error-correcting codes,
| spatiel orientation in robotics, GPS calculations, etc.
|
| https://www.physicsforums.com/threads/linear-algebra-a-moder...
| HybridCurve wrote:
| If you think a textbook is good, the digital edition should
| available for review as they will often be invaluable in hardcopy
| for continued reference. I am pleased the author adheres in some
| part to this mode of thinking.
| Buttons840 wrote:
| I've been waiting for this for 6th months. Thanks to Sheldon
| Axler for making it available for free. This is intended to be a
| _second_ book on Linear Algebra.
|
| For a _first_ book I suggest "Linear Algebra: Theory, Intuition,
| Code" by Mike X Cohen. It's a bit different than a typical math
| textbook, it has more focus on conversational explanations using
| words, although the book does have plenty of proofs as well. The
| book also has a lot of code examples, which I didn't do, but I
| did appreciate the discussions related to computing; for example,
| the book explains that several calculations that can be done by
| hand are numerically unstable when done on computers (those darn
| floats are tricky). For the HN crowd, this is the right focus,
| math for the sake of computing, rather than math for the sake of
| math.
|
| One insight I gained from the book was the 4 different
| perspectives of matrix multiplication. I had never encountered
| this, not even in the oft-suggested "Essence of Linear Algebra"
| YouTube series. Everything I had seen explained only one of the 4
| views, and then I'd encounter a calculation that was better
| understood by another view and would be confused. It still bends
| my mind to think all these different perspectives describe the
| _same calculation_ , they're just different ways of interpreting
| it.
|
| At the risk of spamming a bit, I'll put my notes here, because
| this is something I've never seen written down elsewhere. The
| book has more explanation, these are just my condensed notes.
|
| 4 perspectives on matrix multiplication
|
| =======================================
|
| 1 Element perspective (all possible dot / inner products)
| ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
| (row count x row length) x (column length x column count)
| In AB, every element is the dot product of the corresponding row
| of A and column of B. The rows in A are the same
| length as the columns in B and thus have dot products.
|
| 2 Layer perspective (sum of outer product layers)
| ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
| (column length x column count) x (row count x row length)
| AB is the sum of every outer product between the corresponding
| columns in A and rows in B. The column count in
| A is the same as the row count in B, thus the columns and
| rows pair up exactly for the outer product operation. The
| outer product does not require vectors to be the same length.
|
| 3 Column perspective (weighted sums / linear combinations)
| ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
| (column length x column count) x (column length x column count)
| In AB, every column is a weighted sum of the columns in A; the
| weights come from the columns in B. The weight
| count in the columns of B must match the column count in A.
|
| 4 Row perspective (weighted sums / linear combinations)
| ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
| (row count x row length) x (row count x row length) In
| AB, every row is a weighted sum of the rows in B; the weights
| come from the rows in A. The weight count in the
| rows of A must match the row count in B.
| ndriscoll wrote:
| The most important interpretation IMO is that a matrix is a
| specification for a linear map. A linear map is determined by
| what it does to a basis, and the columns of a matrix are just
| the list of outputs for each basis element (e.g. the first
| column is `f(b_1)`. The nth column is `f(b_n)`). If A is the
| matrix for f and B the matrix for g (for some chosen bases),
| then BA is the matrix for the composition x -> g(f(x)). i.e.
| the nth column is `g(f(b_n))`.
|
| The codomain of f has to match the domain of g for composition
| to make sense, which means dimensions have to match (i.e. row
| count of A must be column count of B).
| mohamez wrote:
| Linear Algebra Done Right is a good book for people who want to
| study the subject of linear algebra in a proof based,
| mathematically rigorous way.
|
| Here [1] you can find Sheldon Axler himself explaining the topics
| of the book in his YouTube channel! How wonderful is that!
|
| Here [2] you can find the solutions to the exercises in the book.
|
| [1]
| https://www.youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmS...
|
| [2] http://linearalgebras.com/
| ayakang31415 wrote:
| This is a great book, but you must first be familiar with proof-
| focused mathematics (logic and set theory). If you are not, I
| first suggest you study a book called "How to Prove It: A
| Structured Approach" by Daniel J. Velleman before studying LA
| Done Right.
| noqc wrote:
| You need a setting in which to learn proof based mathematics,
| and linear algebra really is the first place where students are
| ready for that journey. Not everyone is going to be able to do
| it, but it's very incorrect to say that you must be familiar
| with proof. One must start somewhere, and ZF ain't it.
| ayakang31415 wrote:
| Sorry, what is ZF?
| ekm2 wrote:
| This is good as a second course on Linear Algebra.For a first
| course,use (I am not kidding) _Linear Algebra Done Wrong_ by
| Sergei Treil
|
| https://www.math.brown.edu/streil/papers/LADW/LADW.html
| endymi0n wrote:
| omg just had a look and this one is just everything I hate
| about mathematics and academia.
|
| Starts with lots of random definitions, remarks, axioms and
| introducing new sign language while completely disregarding
| introducing what it's supposed to do, explain or help with.
|
| All self-aggrandization by creating complexity, zero intuition
| and simplification. Isn't there anybody close to the Feynman of
| Linear Algebra?
| PartiallyTyped wrote:
| What about Axler's then?
| gadrev wrote:
| Gilbert Strang's course on Linear Algebra. Playlist:
| https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D
|
| Not as big in scope, though, but great introduction.
| ekm2 wrote:
| There is no royal road bro
| 77pt77 wrote:
| In his mind his above royals, so a royal road would be
| demeaning to him.
| bscphil wrote:
| Yeah, a good example is on the second page of the first
| chapter:
|
| > Remark. It is easy to prove that zero vector 0 is unique,
| and that given v [?] V its additive inverse -v is also
| unique.
|
| The is the first time the word "unique" is used in the text.
| Students are going to have no idea whether this is meant in
| some technical sense or just conventional English. One can
| _imagine_ various meanings, but that doesn 't substitute for
| real understanding.
|
| This is actually why I feel that mathematical texts tend to
| be _not rigorous enough_ , rather than too rigorous. On the
| surface the opposite is true - you complain, for instance,
| that the text jumps immediately into using technical language
| without any prior introduction or intuition building. My take
| is that intuition building doesn't need to replace or preface
| the use of formal precision, but that what is needed is to
| bridge concepts the student already understands and has
| intuition for to the new concept that the student is to
| learn.
|
| In terms of intuition building, I think it's probably best to
| introduce vectors via talking about Euclidean space - which
| gives the student the possibility of using their _physical_
| intuitions. The student should build intuition for _how_ and
| _why_ vector space "axioms" hold by learning that
| fundamental operations like addition (which they already
| grasp) are being extended to vectors in Euclidean space. They
| _already_ instinctively understand the axiomatic properties
| being introduced, it 's just that the raw technical language
| being thrown at them fails to connect to any concept they
| already possess.
| newprint wrote:
| > Remark. It is easy to prove that zero vector 0 is unique,
| and that given v [?] V its additive inverse -v is also
| unique.
|
| I'm sorry, this book is meant for the audience who can read
| and write proofs. Uniqueness proofs are staple of
| mathematics. If word "unique" throws you off, then this
| book is not meant for you.
| curiousgal wrote:
| No offense to OP but you are right. I get the feeling
| that people keep looking for a math-free resource to
| learn math...
| CalChris wrote:
| Now _that_ would be unique.
| wtallis wrote:
| I'd go a bit further and say that if you're not
| comfortable with the basics of mathematical proofs, then
| you're not ready for the subject of linear algebra
| regardless of what book or course you're trying to learn
| from. The purely computational approach to mathematics
| used up through high school (with the oddball exception
| of Euclidean geometry) and many introductory calculus
| classes can't really go much further than that.
| BoiledCabbage wrote:
| > This is actually why I feel that mathematical texts tend
| to be not rigorous enough, rather than too rigorous.
|
| The thing that mathematicians refuse to admit is that they
| are _extremely_ sloppy with their notation, terminology and
| rigor. Especially in comparison to the average programmer.
|
| They are conceptually/abstractly rigorous, but in
| "implementation" are incredibly sloppy. But they've been in
| that world so long they can't really see it / just expect
| it.
|
| And if you debate with one long enough, they'll eventually
| concede and say something along the lines of "well math
| evolved being written on paper and conciseness was
| important so that took priority over those other concerns."
| And it leaks through into math instruction and general math
| text writing.
|
| Programming is forced to be extremely rigorous at the
| implementation level simply because what is written must be
| executed. Now engineering abstraction is extremely
| conceptually sloppy and if it works it's often deemed "good
| enough". And math generally is the exact opposite. Even for
| a simple case, take the number of symbols that have context
| sensitive meanings and mathematicians. They will use them
| without declaring which context they are using, and a
| reader is simply supposed to infer correctly. It's actually
| somewhat funny because it's not at all how they see
| themselves.
| rq1 wrote:
| This is ridiculous.
|
| The average computer scientist (not only "programmer", as
| a js dev would be) never wrote lean/coq or similar, and
| is not aware of the Curry-Haskell like theorems and their
| implications.
| aidos wrote:
| 3Blue1Browns Essence of Linear Algebra is my go to
|
| https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x.
| ..
| resource0x wrote:
| > Isn't there anybody close to the Feynman of Linear Algebra?
|
| No. The subject is too young (the first book dedicated to
| Linear Algebra was written in 1942). Since then, there have
| been at least 3 generations of textbooks (the first one was
| all about matrices and determinants). That was boring. Each
| subsequent iteration is worse.
|
| What is dual space? What motivates the definition? How useful
| is the concept? After watching no less than 10 lectures on
| the subject on youtube, I'm more confused than ever.
|
| Why should I care about different forms of matrix
| decomposition? What do they buy me? (It turns out, some of
| them are useful in computer algebra, but the math textbook is
| mum about it)
|
| My overall impression is: the subject is not well understood.
| Give it another 100 years. :-)
| rq1 wrote:
| Why should it buy you something is the real question.
|
| You don't need to understand it the way the "initial"
| author thought about it, should that person had given it
| more thoughts...
|
| History of maths is really interesting but it's not to be
| confused with math.
|
| Concepts are not useful as you think about them in economic
| opportunity case. Think about them as "did you notice that
| property" and then you start doing math, by playing with
| these concepts.
|
| Otherwise you'll be tied to someones way of thinking
| instead of hacking into it.
| bumbledraven wrote:
| I like the free course on linear algebra by Strang's Ph.D
| student Pavel Grinfeld. It's a series of short videos with
| online graded exercises. Most concepts are introduced using
| geometric vectors, polynomials, and vectors in Rn as
| examples. https://www.lem.ma/books/AIApowDnjlDDQrp-
| uOZVow/landing
| tenderfault wrote:
| Right. "Complex numbers were invented so that we can take square
| roots of negative numbers"
|
| Why? I wish a textbook tell me why, right there.
| jjoonathan wrote:
| Because complex numbers make the fundamental theorem of Algebra
| nice and simple rather than complicated and ugly. In turn, this
| makes the spectral theorem of linear algebra nice and simple
| rather than complicated and ugly. In turn, this makes a bunch
| of downstream applications nice and simple rather than
| complicated and ugly.
|
| You will get a feel for this if you work Axler's problems. More
| importantly, you will gain an intuition for the fact that if
| you turn up your nose at complex numbers while going into these
| application spaces, you are likely to painstakingly reinvent
| them except harder, more ugly, and worse.
|
| Example: in physics, oscillation and waves A. underpin
| everything and B. involve energy sloshing between _two_
| buckets. Kinetic and potential. Electric and magnetic. Pressure
| and velocity. These become real and imaginary (or imaginary and
| real, it 's arbitrary). This is where complex numbers -- where
| you have two choices of units -- absolutely shine. Where you
| would have needed two coupled equations with lots of sin(),
| cos(), trig identities, and perhaps even bifurcated domains you
| now have one simple equation with exponentials and lots of
| mathematical power tools immediately available. Complex numbers
| are a _huge_ upgrade, and that 's why anything to do with waves
| will have them absolutely everywhere.
| wwarner wrote:
| You might think that in every real world application, complex
| numbers are introduced as a convenience, and that every
| calculation that takes advantage of them ends with taking the
| real part of the result, but that's not the case. In QM, the
| final answer contains an imaginary part that cannot be
| removed.
| akoboldfrying wrote:
| I think it's the same reason why negative numbers were
| invented: It lets you do more with algebra than you could
| before (some of which, like raising something to a power
| leading to a sine wave is pretty weird, but turns out to be
| useful in engineering, etc.), and everything else still "just
| works" the same way as before.
|
| (Admittedly the applications of negative numbers are much more
| obvious.)
| fuzztester wrote:
| And why zero was invented.
| fuzztester wrote:
| Because mathematicians like to make up things and theories to
| feel important, since they are impractical people who don't do
| anything important in the real world.
|
| Half-joke apart (and I studied math in college, BTW, as my
| major, with Sanskrit as a minor), complex numbers have many
| uses in the real world, in engineering and other areas.
|
| See the Applications section of
| https://en.m.wikipedia.org/wiki/Complex_number
| fuzztester wrote:
| Mathematicians are _infinitely_ better than statisticians,
| though, because the definition of a statistician is "a
| person who can have his head in an oven and his feet in a
| freezer", and say, "on the average, I am feeling quite
| comfortable".
| fuzztester wrote:
| Tons of things and phenomena in the _real_ world are based on
| mathematics. Plant and leaf patterns, ocean waves, water
| flowing in tubes or channels, the weather, mineral and plant
| and animal structures, rain and snow and ice, mountains,
| deserts, glaciers, floods, thunder and lightning,
| electromagnetism, fire, etc., etc., etc. And some of those
| things are _really_ based on _imaginary_ numbers.
|
| ;-)
| fiforpg wrote:
| Complex numbers are algebraically closed, reals are not. This
| means, if you write a polynomial with complex coefficients, it
| will have (only) complex roots. Analogous statement for the
| reals isn't true.
| madvoid wrote:
| Imagine you're a 16th century Italian mathematician who is
| trying to solve cubic equations. You notice that when you try
| to solve some equations, you end up with a sqrt(-1) in your
| work. If you're Cardano, you call those terms "irreducible" and
| forget about them. If you're Bombelli, you realize that if you
| continue working at the equation while assuming sqrt(-1) is a
| distinct mathematical entity, you can find the real roots of
| cubic equations.
|
| So I would say that it's less that "Complex numbers were
| invented so that we can take square roots of negative numbers",
| and more "Assuming that sqrt(-1) is a mathematical entity lets
| us solve certain cubic equations, and that's useful and
| interesting". Eventually, people just called sqrt(-1) "i", and
| then invented/discovered a lot of other math.
|
| Source:
| http://fermatslasttheorem.blogspot.com/2006/12/bombelli-and-...
| Nevermark wrote:
| I prefer a simpler perspective for complex numbers of "defined
| latently, then discovered, accepted, named and given notation",
| other than "invented".
|
| Invented implies some degree of arbitrariness or choice, but
| complex numbers are not an arbitrary construct.
|
| Zero, negative numbers, and imaginary numbers were all latently
| defined by prior concepts before they were recognized. They
| were unavoidable, as existing operations inevitably kept
| producing them. Since they kept coming up, it forced people to
| eventually recognize that these seemingly nonsensical concepts
| continued to behave sensibly under the operations that produced
| them.
|
| Once addition and subtraction were defined on natural numbers,
| (1, 2, 3, ... etc), the concept of zero was latently defined.
| The concept of "nothing" was not immediately recognized as a
| number, but there is only one consistent way of dealing with
| 2-2, 5-5, 7-7, etc. Eventually that concept was given a name
| "zero", notation "0", and adopted as a number.
|
| It was discovered, in that it was already determined by
| addition and subtraction, just not yet recognized.
|
| Similarly with negative numbers. They were also latently
| determined by addition and subtraction. At first subtracting a
| larger number from a smaller number was considered nonsensical.
| But starting from the simple acceptance that "5-8" can at least
| be consistently viewed as the number which added to 8 gives 5,
| and other similar examples, it was discovered that such numbers
| had only one consistent behavior.
|
| So they were accepted, given a name "negative numbers" and a
| notation "-x", short hand for "0-x".
|
| And again, once addition, multiplication, (and optionally
| exponentiation) were defined, the expressions x*x = -1 (or x =
| sqrt(-1)) were run into, they were initially considered non-
| sensical.
|
| But starting from acceptance that it at least makes sense to
| say that "the square of the square root of -1", is "-1", it was
| discovered that roots of -1 could be worked with consistently
| using the already accepted operations that produced them.
|
| The numbers that included square roots of -1 were given a name
| "imaginary numbers", the square root of -1 given notation, "i",
| and we got complex numbers that had both real and square root
| of -1 parts.
| Ericson2314 wrote:
| At least it gave you one round of a "why" answer!
| MrBlueIncognito wrote:
| I don't know if it's just me, but I'm terribly lost in the search
| for the right texts. Every time I come across a new
| book/resource, it compounds the confusion. I find myself
| incapable of sitting down with a book and working through it
| without switching to another book in-between. I'd be really happy
| to hear if anyone has found a solution to this unproductive but
| sticky habit.
| generationP wrote:
| 4th edition and still hasn't bothered to disambiguate between
| polynomial and polynomial function.
|
| Still a good example of mathematical writing. But, as Einstein
| supposedly said, "as simple as possible but not simpler". Why is
| it always American authors that forget that last part?
| slowhadoken wrote:
| the good old indispensable and underappreciated linear algebra
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