[HN Gopher] A 2020 Vision of Linear Algebra
___________________________________________________________________
A 2020 Vision of Linear Algebra
Author : organicfigs
Score : 682 points
Date : 2020-05-12 04:55 UTC (14 hours ago)
(HTM) web link (ocw.mit.edu)
(TXT) w3m dump (ocw.mit.edu)
| elAhmo wrote:
| What would you recommend as a good resource for learning about
| Linear Algebra in 2020?
|
| I am aware of his course on OCW, but wondering is there something
| more interactive and/or newer than those lectures that has
| similar quality.
| skywal_l wrote:
| Frankly, couple with this book, it does hardly get better. You
| still have 3blue1brown[1] series of video, but it just brush
| off the surface.
|
| [1]
| https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...
| crdrost wrote:
| So I like this outline. It is very MIT-ish where there is a sense
| of teaching someone to solve practical engineering problems with
| matrices.
|
| But, I do foresee some difficulties. One thing that I find really
| difficult, for example, is that I take undergrads who have had
| linear algebra and ask "what is the determinant?" and seldom get
| back the "best" conceptual answer, "the determinant is the
| product of the eigenvalues." Like, this is math, the best answer
| should not be the only one, but it should be ideally the most
| popular. We would consider it a failure in my mind if the most
| popular explanation of the fundamental theorem of calculus was
| not some variation of "integrals undo derivatives and vice
| versa". I don't see this approach solving that. Furthermore there
| is a lot of focus from day one on this CR decomposition which
| serves to say that a linear transform from R^m to R^n might map
| to a subspace of R^n with smaller dimension r < min(m,n) and
| while in some sense this is true it is itself quite "unphysical"
| --if a matrix contains noisy entries then it will generally only
| be degenerate in this way with probability zero. (You need
| perfect noise cancelation to get degeneracies, which amounts to a
| sort of neglected underlying conserved quantity which is pushing
| back on you and demanding to be conserved.) In that sense the CR
| decomposition is kind of pointless and is just working around
| some "perfect little counterexamples". So it seems weird to see
| someone say "hold this up as the most important thing!!"
| ABeeSea wrote:
| The most intuitive definition of determinant for me is it's the
| growth or shrink factor of a differentiable function at a
| point. The eigenvalues of the Jacobian just demonstrate the
| various sources and sinks in a dynamic system.
|
| This view also motivates the concept of vector bundles and
| vector spaces at a point.
| mansoor_ wrote:
| Subjective, I find the geometric interpretation of the
| determinant to be the "best".
| heinrichhartman wrote:
| > seldom get back the "best" conceptual answer, "the
| determinant is the product of the eigenvalues."
|
| I found that the "best conceptual" answer depends a lot on
| taste, and what concepts you are familiar with.
|
| In this case:
|
| - Calculating exact eigenvalues of matrices larger than 4x4 is
| impractical, since it requires you to solve a polynomial of
| degree >4.
|
| - The EV exist only in algebraically closed fields (complex
| numbers), while the determinant itself lives in the base field
| (rationals, reals).
|
| How about:
|
| - [Geometric Determinant] The determinant is the volume of the
| polytope (parallel-epiped) spanned by the column vectors of the
| matrix.
|
| - [Coordinate Free Determinant] The determinant is the map
| induced between the highest exterior powers of the source and
| target vector spaces
| (https://en.wikipedia.org/wiki/Exterior_algebra)
|
| - I think there is also a representation theoretic version,
| that characterizes the determinant as invariant under the
| Symmetric group acting by permutation on the columns/rows of
| the matrix.
| Bootvis wrote:
| I don't have an intuition for these concepts I'm afraid (I
| probably should watch the videos). What I don't see for
| instance is how this relates to the fact that a matrix A with
| det(A) = 0 is not invertible.
| crdrost wrote:
| So an eigenvector is a special direction which most linear
| transformations have. If you ask them to transform a vector
| in that direction, they do not rotate that direction to
| some other direction, they only scale it by a constant
| called the eigenvalue. Usually there are a bunch of these
| directions for one transform, and they do not need to be
| orthogonal. We often choose one vector as representative,
| like (-1, 1): but this is shorthand for all vectors (-t, t)
| for all t. One important thing is that the zero vector (0,
| 0) doesn't count (even though T 0 = 0) because it can't
| represent a whole direction.
|
| So for example if I take (x, y) to T (x, y) = (3x + y, 2x +
| 4y), that is an example of what we call a linear
| transformation -- it obeys T(p1 + p2) = T p1 + T p2, it
| distributes over addition.
|
| Now in addition to noticing that this is linear we may
| happen to notice that T (-1, 1) = (-3 + 1, -2 + 4) = (-2,
| 2). So in the direction (-t, t) we are just scaling vectors
| by a factor of 2, to (-2t, 2t). Similarly we might notice
| that T (1, 2) = (3 + 2, 2 + 8) = (5, 10). So in the
| direction (t, 2t) we are just scaling vectors by a factor
| of 5 to (5t, 10t).
|
| These two scaling factors, 2 and 5, are called the
| eigenvalues of T. Their product, 10, is called the
| determinant of T. And in this case their eigenvectors span
| the entire space -- you can make any other (a, b) as a
| combination (-t1, t1) + (t2, 2 t2), for some numbers t1,
| t2. Actually t1 = (-2a + b)/3 and t2 = (a + b)/3, I can
| work out pretty quickly. And in this t-space this
| transformation is very easy to think about, it has been
| "diagonalized."
|
| Sometimes these eigenvalues and eigenvectors don't exist,
| but we can patch that up with one of two tricks. The first
| trick is, for example, used for the 2x2 rotation matrices.
| These rotate every direction into some other direction, so
| how will I find some direction which "stands still"? The
| answer here is complex numbers, in this case it turns out
| that any 2x2 rotation by angle t will map the complex
| vector (1, i) to (cos t + i sin t, -sin t + i cos t) = (cos
| t + i sin t) * (1, i), so it has two complex eigenvalues
| e^(it), e^(-it). So the first trick is complex numbers.
| There is, it turns out, only one other class of weird
| transformation. In these weird transformations, it is
| possible to define chains of "generalized eigenvectors".
| Each chain starts with one ordinary eigenvector with an
| ordinary eigenvalue q, T v1 = q v1, and then the next
| element of the chain is a "generalized eigenvector of rank
| 2" which has T v2 = v1 + q v2, and then the next element of
| the chain is a "generalized eigenvector of rank 3" which
| has T v3 = v2 + q v3, and so on.
|
| So it is a theorem that any NxN complex linear
| transformation has N linearly independent generalized
| eigenvectors which span the space, and "usually" these are
| all just normal eigenvectors and the matrix is
| "diagonalizable" (and even if they aren't, they come in
| families which start from one normal eigenvector and the
| matrix can be put into "Jordan normal form").
|
| If you understood all of that, you are ready for the main
| result that you asked about. :)
|
| For a linear transformation to be invertible, it needs to
| map distinct input vectors to distinct output vectors. If
| it maps two different input vectors to the same output,
| then invertibility fails.
|
| So we know that invertibility fails when we can find
| distinct v1 and v2 such that T v1 = T v2.
|
| Put another way, T v1 - T v2 = 0. But by the linearity
| property, T distributes over additions and subtractions, so
| this is the same as saying that T (v1 - v2) = 0 for v1 - v2
| nonzero. This is enough to establish that v1 - v2 is an
| eigenvector with eigenvalue zero.
|
| What does this do to the determinant, the product of all
| the eigenvalues? Well, zero times anything is zero. So if
| some linear transformation T is not invertible, then you
| immediately can conclude that det(T) = 0.
|
| Furthermore this argument goes the other way too, with only
| a little subtlety related to these "generalized
| eigenvalues" -- basically, that the generalized eigenvalues
| always exist and there is always at least one eigenvector
| which actually has that eigenvalue, and that complex
| numbers still have this property that any finite product of
| complex numbers which results in zero can only come about
| if one of those numbers was zero. If you know all of those
| things, then you can work your way backwards to conclude
| that det(T) = 0 implies that one of the generalized
| eigenvalues is zero, which has at least one normal
| eigenvector v such that T v = 0, which I can then use to
| find many inputs for any given output, T u = T (u + k v)
| for any k
|
| So to say that this product-of-eigenvalues is zero is to
| say that one of the eigenvalues is zero, and therefore the
| linear transformation is projecting down to some smaller,
| flatter subspace in a way that cannot be uniquely undone.
| If there is no such smaller flatter subspace, then the
| transformation must have been invertible all along.
| heinrichhartman wrote:
| The geometric version is the most intuitive for me:
|
| If the volume of the prallel-epiped is zero, then there
| will be directions in the target space, that you did not
| hit. Hence he matrix can not be invertible.
| Bootvis wrote:
| I will have to understand what directions in target space
| are first. I'll guess I'll have to do the work ;)
| vlasev wrote:
| Take a 3x3 matrix A for example. Then det(A) is the volume
| of the parallepiped formed by the row vectors. If one
| vector is a linear combination of the other two, this means
| that the vectors lie in a plane, which has volume 0 in 3D,
| so det(A) = 0. Since we have a plane in 3D, this means A
| can't express all vectors in 3D, so it's not invertible.
| This generalizes to any dimension.
| abdullahkhalids wrote:
| Re: your last point, the determinant is the matrix function,
| which is fully anti-symmetric under permutation of rows and
| columns i.e. swapping a pair of rows or a pair of cols pulls
| out a minus sign. The definition of the determinant is
| related to the alternating representation of the symmetric
| group.
|
| The permanant [1] is the matrix function which is fully
| symmetric, so permuting any rows or cols leaves it invariant.
| It emerges from the identity representation.
|
| Finally, partially symmetric matrix functions are known as
| immanants [2], defined using the other irreps of the
| symmetric group.
|
| [1] https://en.wikipedia.org/wiki/Permanent_%28mathematics%29
|
| [2] https://en.wikipedia.org/wiki/Immanant
| frequentnapper wrote:
| Back in uni (2005), we used Dr. Strang's text for linear algebra.
| When reading the text, I felt like some down-to-earth professor
| was trying to explain these difficult topics as simply as
| possible. I remember discovering mit.edu back then and finding
| precious video lectures that went along with the book after the
| course. One of the very few times I was so genuinely happy and
| excited to watch math lectures online :p
| jbd28 wrote:
| We used his book then too, at Drexel in Philadelphia. Our prof
| at the time invited Dr Strang to guest lecture one time and I
| remember it being so clear and obvious as he talked that I
| thought "wow this is why an MIT education is so revered".
|
| I waited after the lecture to personally thank him and have him
| autograph the textbook; very glad I did in retrospect.
| balls187 wrote:
| Admittedly, I never fully groked linear algebra.
|
| Some of the concepts made sense, especially solving for linear
| systems of equations.
|
| Recently, I decided to brush up on my math skills via Youtube
| videos, and came across this series:
| https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
|
| It explains Linear Algebra concepts using 2D and 3D vector
| manipulation, and the animations help me visualize the underlying
| maths.
| roenxi wrote:
| It is interesting to compare this with 3Blue1Brown's linear
| algebra introduction on YouTube. He seems to have been the only
| mathematician who has actually mastered the medium; linear
| algebra lends itself very well to animations. The mathematicians
| don't understand how badly they need to animate some of these
| concepts.
| gfxgirl wrote:
| 3Blue1Brown's linear algebra animations were fun to watch but
| they did almost nothing for me except the basic fact that the
| "linear" part means lines.
|
| The rest was effectively preaching to the choir so those that
| already know linear algebra nodded their heads and idiots like
| me were still flummoxed
| john4532452 wrote:
| Yes. 3Blue1Brown has himself said multiple times no videos is
| substitute for text books.
| domnomnom wrote:
| Such is life
| mjburgess wrote:
| That's interesting. I do think 3B1B's goal is probably to
| build better intuitions in people who already know it.
| mesaframe wrote:
| Yes, those lectures are good enough for intuitions only.
| For practice one has to read books. I just hope the viewer
| knows that. Which I have seen is absent among some people.
| ekianjo wrote:
| Agree, world class education in maths there. He understands the
| importance of examples and visualization and it changes
| everything.
| gowld wrote:
| Mathematicians and teachers aren't all computer programmers, so
| creating animations isn't their forte.
|
| Books have pictures that do a pretty good job.
|
| Animations are pretty and interestit but that isn't the same as
| teaching all the math.
| [deleted]
| auggierose wrote:
| Have not watched the videos yet, but that seems to me more like
| an 1820 vision of linear algebra :-)
|
| If you look at the order of topics in his book "An Introduction
| to Linear Algebra", you will find the topic "Linear
| Transformation" way back in chapter 8! Even after the chapters
| eigenvalue decomposition and singular value decomposition. But
| understanding that a matrix is just the representation of a
| linear transformation in a particular basis is probably the most
| important and first thing you should learn about matrices ...
| JoeCamel wrote:
| I also think that this point should be emphasized more. It
| helped me a lot when I realized it. I also liked the abstract
| approach to vector spaces. Of course, a matrix is just one way
| to represent linear transformations and it can also represent
| other things like a system of linear equations.
| plandis wrote:
| When I was self studying linear algebra I found Strange's book
| to be inadequate on its own, you really need the lectures to
| get the most value out of it.
|
| I found going through Linear Algebra Done Right to provide a
| good counterbalance to Strang's book+lectures.
| tchaffee wrote:
| > Have not watched the videos yet,
|
| Then please do. I took several online linear algebra courses
| from sources I trust and they were pretty bad. Or let's put it
| another way: I'm a pretty clever guy and I was still left
| confused. Strang is excellent in the classroom, and I almost
| even like videos for learning now thanks to him (x1.5 speed is
| your friend). His videos should not be your only learning
| source, but judging his course only by the book might result in
| a lot of learners skipping what I found to be the best course
| by far. If you want to learn linear algebra, give Strang a try
| first and you might save a lot of time.
| [deleted]
| gowld wrote:
| This is silly. LA is incredibly rich, broad, and feel area of
| study. You can't just grab one part of it and say it's the most
| important and first thing. And it's silly to say that whatever
| is most important should be first -- the central ideas depend
| on prepaeatt.
| bo1024 wrote:
| I'm not sure I agree when that one thing is the word linear
| (half the name). It just feels wrong not to ground the
| subject in the idea of a linear transformation.
| dandanua wrote:
| I don't think it was this way in 1820, but I agree with your
| point.
|
| Even though I also use Linear Algebra mostly computationally
| today, the origin of it is in the geometry and I think this
| connection should come first. Also, "number crunching" is a
| boring way to learn things.
|
| Though, "matrix way" can be good for engineers.
| jacobolus wrote:
| Gaussian Elimination is indeed from the 1820s. All the rest is
| more recent than that. The idea of matrix decomposition per se
| comes from the 1850s. The earliest work on something like the
| SVD is from the 1870s.
|
| You are onto something though. Strang is coming from a
| direction of numerical computations and algorithms for solving
| real-world problems. Pure mathematics departments for at least
| the past maybe 80 years often look down on numerical analysis,
| statistics, engineering, and natural science, and adopt a
| position that education of students should be optimized in the
| direction of helping them prove the maximally general results
| using the most abstract and technical machinery, with an
| unfortunate emphasis on symbol twiddling vs. examining concrete
| examples. By contrast, in the 19th century there was much more
| of a unified vision and more respect for computations and real-
| world problems. Gauss himself was employed throughout his
| career as an astronomer / geodesist, rather than as a
| mathematician, and arguably his most important work was
| inventing the method of least squares, which he used for
| interpreting astronomical observations.
|
| With the rise of electronic computers, it is possible that the
| dominant 2050 vision of linear algebra and the dominant 1900
| vision of linear algebra will be closer to each-other than
| either one is to a 1950 vision from a graduate course in a pure
| math department.
| mattkrause wrote:
| Indeed, Strang's textbook starts with "I believe the teaching
| of linear algebra has become too abstract."
|
| He mentions this sentiment in a lot of interviews and things
| too.
| BeetleB wrote:
| If you want to get into serious physics/engineering, the
| abstract aspect of linear algebra is _much_ more important
| than the boring computational mechanics. And quite a bit of
| those computational mechanics lead you astray when you go to
| infinite dimensions.
| atrettel wrote:
| > Pure mathematics departments for at least the past maybe 80
| years often look down on numerical analysis, statistics,
| engineering, and natural science, and adopt a position that
| education of students should be optimized in the direction of
| helping them prove the maximally general results using the
| most abstract and technical machinery, with an unfortunate
| emphasis on symbol twiddling vs. examining concrete examples.
|
| I had this view when I took linear algebra as an
| undergraduate, but I have gradually changed on the subject
| over time. I took a standard "linear algebra for scientists
| and engineers" course but I found it too abstract at the
| time. The instructor rarely concentrated on examples and
| applications despite the more applied focus in the course
| title. Later I came to appreciate the abstraction, since it
| helped me understand more advanced mathematical topics
| unrelated to the "number-crunching" I originally associated
| the topic with. I now think the instructor had a more
| "unified" approach, but I didn't realize it at the time.
| auggierose wrote:
| I believe in applications and theory going hand in hand
| together and benefiting each other. The computer is an
| incredibly powerful tool perfectly suited for this purpose.
| If we resist the urge to just see it as a push-button
| technology. Viewing matrices as a box of numbers instead of
| as a representation of a linear transformation leans too much
| in the direction of push-button for my taste.
| jacobolus wrote:
| Gil Strang does not view matrices as "just boxes of
| numbers", nor does he teach that view.
|
| YMMV, but I find pure mathematicians treat computers as
| "push-button technology" much more than applied
| mathematicians.
| auggierose wrote:
| I am not disagreeing with you there when it comes to pure
| mathematicians :-D
|
| Edit: But there are of course big exceptions there as
| well, for example Thomas Hales.
| vlasev wrote:
| I believe this is largely because in the field of
| mathematics, Linear algebra is just the seed that sprouts the
| growth of other very very useful mathematical subjects like
| Abstract Algebra, Functional Analysis and so on. Linear
| Algebra is used as a stepping stone to more general theories
| that are also super useful.
|
| Take Hilbert spaces for example. They are based on linear
| algebra. They are quite general and you might argue that
| there's a lot of symbol twiddling there. However, Hilbert
| spaces are/were essential in the study of Quantum Mechanics,
| which we can argue is a very important topic.
|
| And if you only stick with matrices and numerics, you're
| bound to get stuck in the numbers and details and miss the
| big picture. A lot of results are much cleaner to obtain once
| you divorce yourself from the concrete world of matrix
| representation.
|
| Of course, we should probably have the best of both worlds.
| I'm not saying applications are unimportant. Take something
| like signal processing, which relies heavily on both numerics
| and general theory.
|
| So I'd like to add something to your point. Math departments
| optimize the education of math students towards the more
| general, and perhaps students not interested in pursuing pure
| math should have course-work that reflects that.
| Koshkin wrote:
| > _a matrix is just the representation of a linear
| transformation_
|
| While this view certainly helps intuition at initial stages of
| learning, it is not "just" that, and computational methods
| involving matrices are of much more practical importance
| (similar to being able to add and multiply numbers which we are
| taught early in life) which is probably why the stress is on
| them first and foremost. Someone said, "learn to calculate,
| understanding will come later."
| ivan_ah wrote:
| For anyone who is already familiar with the Prof. Strang's
| lectures from previous years, the main new thing in this five-
| lecture mini-series is he tries to condense the material even
| further--maximum intuition and power-ideas, instead of the full-
| length in-class lecture format with derivations. This makes the
| material difficult to understand for beginners, but makes a great
| second source in addition to or after a regular LA class.
|
| One of the interesting new ways of thinking in these lectures is
| the A = CR decomposition for any matrix A, where C is a matrix
| that contains a basis for the column space of A, while R contains
| the non-zero rows in RREF(A) -- in other words a basis for the
| row space, see
| https://ocw.mit.edu/resources/res-18-010-a-2020-vision-of-li...
|
| Example you can play with:
| https://live.sympy.org/?evaluate=C%20%3D%20Matrix(%5B%5B1%2C...
|
| Thinking of A as CR might be a little intense as first-contact
| with linear algebra, but I think it contains the "essence" of
| what is going on, and could potentially set the stage for when
| these concepts are explained (normally much later in a linear
| algebra course). Also, I think the "A=CR picture" is a nice
| justification for where RREF(A) comes about... otherwise students
| always complain that the first few chapters on Gauss-Jordan
| elimination is "mind-numbing arithmetic" (which is kind of
| true...) but maybe if we present the algorithm as "finding the
| CR-decomposition which will help you understand dozens of other
| concepts in the remainder of the course" it would motivate more
| people to learn about RREFs and the G-J algo.
| ivan_ah wrote:
| Since y'all are code-literate, here the SymPy function for
| finding the CR-decomposition of any matrix A:
| def crd(A): """ Computes the CR
| decomposition of the matrix A. """ rrefA,
| licols = A.rref() # compute RREF(A) C = A[:, licols]
| # linearly indep. cols of A r = len(licols)
| # = rank(A) R = rrefA[0:r, :] # non-zero rows
| in RREF(A) return C, R
|
| Test to check it works:
| https://live.sympy.org/?evaluate=A%20%3D%20Matrix(%5B%0A%20%...
| tomerbd wrote:
| It was a good course I watched it online but I didn't understand
| much.
| inshadows wrote:
| Gilbert Strang taught me how to sanely multiply matrices. His
| Introduction to Linear Algebra is very approachable. It's wildly
| different experience compared to linear algebra courses I had on
| university, it actually makes sense and is fun!
| irl_zebra wrote:
| I've been wanting to learn linear algebra. I had some exposure in
| college along with my calc classes, but never really understood
| it fundamentally. Like it was mentioned, I mostly did matrix
| transforms but didn't realize fundamentally grasp.
|
| I started doing LA on Khan academy, and checked out Linear
| Algebra Done Right. LADR was a little too much into the deep end
| for me. KA seemed to be good. One nice thing about KA is that
| when I didn't quite remember something (i.e. how exactly to
| multiply a matrix) I could just go to an earlier pre-LA lesson,
| pick it up, and then go back to LA where I left off. I'm a few
| lessons in.
|
| What do you all recommend for someone like me?
| nafizh wrote:
| If you want to learn LA by coding, coding the matrix by philip
| klein is a really good book. He even has his Coursera lecture
| videos (not available on coursera anymore last I checked) up on
| his website.
| potta_coffee wrote:
| Picked up Strang's linear algebra book recently and I'm enjoying
| it. I've been consistently impressed with the content of MIT
| books.
| srean wrote:
| I am familiar with the material of linear algebra but haven't
| read his books. Could someone who has absorbed linear algebra
| from different sources and familiar with Strang's books comment
| on what's good and bad and unique about them.
|
| In my time I had picked LA from Ben Noble, Halmos and Axler and
| the computation side of things from Golub & van Loan.
| T-zex wrote:
| I'm still a beginner and I think Strang's Linear algebra books
| are more like a supplement material to his lectures. If you
| need to build a solid theoretical foundation of linear algebra
| you'd need to consider other resources too.
|
| Having said that, he is explaining many things really well and
| is helping a lot to build intuition. He is always cautious
| presenting things that are computationally inefficient and
| suggests the alternatives.
|
| Exercises are too hard for me personally. I'd prefer a more
| laborious set of exercises helping to cement the material, (as
| in calculus or usual algebra) and then have one or two problem
| solving puzzles at the end.
| srean wrote:
| So the focus is on different recipes to cook a matrix with ?
| Different operations one can do on a matrix ?
|
| I hope its not just that, that would be very limiting
| considering what linear algebra is about and capable of.
| T-zex wrote:
| No, his books are not recipes. The thing I'm struggling to
| communicate here is that he's got a more pragmatic style
| compared to other text books. The material he presents is
| complete and and he is doing great job making it
| approachable for non mathematicians.
|
| His books usually expand on the subjects he presents at his
| online lectures. I see them as advanced lecture notes.
| srean wrote:
| No worries, not your fault, its really on me (to read the
| books). Its not really fair of me to ask for a
| comprehensive description in comments. Thanks for your
| comments anyhow.
| mturmon wrote:
| I like your bibliography!
|
| Ben Noble's book was my entry to LA. I was an undergraduate and
| involved in a research activity that demanded a lot of
| knowledge of the eigenvalue problem. The concrete approach in
| that book helped a lot.
|
| It was only later on that I took a class based on G&vL
| (implementing a bunch of basic LA factorizations in Matlab),
| and in my spare time read Halmos's book. I understand the
| coordinate-free algebraic approach, but I work on applications
| and that viewpoint has not stuck with me. The stuff on
| numerical accuracy in GvL really _did_ stick, OTOH.
|
| From the comments here, and Strang's book's table of contents,
| I gather that his book (which has a lot of fans) has a concrete
| geometric approach.
| mturmon wrote:
| Self-reply: Here's a comparative review of Noble+Daniel vs.
| Strang: https://pdf.sciencedirectassets.com/271586/1-s2.0-S00
| 2437950...
| mesaframe wrote:
| I read his book and I would say his book work in complement
| with his lectures. It is not good enough on it's own.
| pengaru wrote:
| It's ridiculous how much random college-level linear algebra
| textbook material I stared at before things clicked in the course
| of just jumping in and exploring 3D graphics and writing my own
| 3D vector, matrix multiplication and 3D transform headers and
| using them in making some games in plain C.
|
| At some point it's like "Wait, is linear algebra really just
| about heaps of multiplication and addition? Like every dimension
| gets multiplied by values for every dimension, and values 0 and 1
| are way more interesting than I previously appreciated. That
| funny identity matrix with the diagonal 1s in a sea of 0s, that's
| just an orthonormal basis where each corresponding dimension's
| axis is getting 100% of the multiplication like a noop. This is
| ridiculously simple yet unlocks an entire new world of
| understanding, why the hell couldn't my textbooks explain it in
| these terms on page 1? FML"
|
| I'm still a noob when it comes to linear algebra and 3D stuff,
| but it feels like all the textbooks in the world couldn't have
| taught me what some hands-on 3D graphics programming impressed
| upon me rather quickly. Maybe my understanding is all wrong, feel
| free to correct me, as my understanding on this subject is
| entirely self-taught.
| mangoman wrote:
| Do you have any recommendations on videos/tutorials with
| graphics stuff that would help me grasp some of the linear
| algebra and basics about 2D/3D graphics? Your comment about
| 'staring at random college-level linear algebra' stuff
| resonated deeply with me, and I've always felt like I'm just
| not understanding how it all connects.
| pengaru wrote:
| Not really, I didn't follow any specific guide. But if you
| like learning from youtube videos Casey Muratori has some
| decent streams about this stuff on his Handmade Hero channel.
| The 3Blue1Brown channel also has some relevant videos.
|
| If you've never written a standalone software-rendered ray
| tracer, I found that to be a very useful exercise early on.
| There are plenty of tutorials for those on the interwebs.
| samvher wrote:
| Not the person you responded to but I found this course to be
| very good: https://www.edx.org/course/computer-graphics-2
| gowld wrote:
| > Wait, is linear algebra really just about heaps of
| multiplication and addition?
|
| That's just one of dozens of things LA is "about"
|
| > why the hell couldn't my textbooks explain it in these terms
| on page 1
|
| Because you wouldn't have understood terms like
|
| > orthonormal
|
| and because it would have been unhelpful to everyone else who
| want in LA for the exact same reason you were.
|
| Being obviously in retrospect doesn't mean it was obvious in
| forespect. You had to learn the material first.
| dxbydt wrote:
| > Maybe my understanding is all wrong, feel free to correct me,
| as my understanding on this subject is entirely self-taught.
|
| I wouldn't say it is all wrong. Just that the stuff you are
| talking about is a very tiny fraction of LA. I took a graduate
| class in LA, based on Strang's book. I have the book right here
| in front of me. So the stuff you allude to, i.e. rotation
| matrix, reflection matrix & projection matrix, is on p130 of
| Chapter 2. We got to that in the 1st month of the semester, &
| it got about 1 hour of classtime total. That's it. An LA class
| is like 4 months, or 50 hours. If the point of LA to derive
| those matrices so one can do 3D computer graphics with scaling,
| rotation & projection ? No, that stuff is too basic. We got 1
| homework problem on that, that's it.
|
| The stuff that most of the class struggled with ( & still
| struggle with, because Strang goes over it rather quickly in
| his book), is function spaces ( chapter 3, p182), Gram Schmidt
| for functions ( p184), FFTs, (p195), fibonacci & lucas numbers
| (p255), the whole stability of differential equations chapter (
| he gives these hard and fast rules like a Differential Equation
| is stable if trace is negative & determinant is positive, but
| its not too clear why. ), quadratic forms & minimum principles
| - that whole 6th chapter glosses over too much material imo.
|
| Overall, Strang's book is a solid A+ on how to get stuff done,
| but maybe a B- on why stuff works the way it works. Like, why
| should I find Rayleigh quotient if I want to minimize one
| quadratic divided by another ? Strang just says, do it & you'll
| get the minimum. How to find a quadratic over [-1,1] that is
| the least distance away from a cubic in that same space ?
| Again, Strang gives a method but the why part of it is quite
| mysterious.
| Phlogistique wrote:
| Is it the same issue as the infamous "monads tutorials"
| problem, where the understanding takes a lot of time to infuse
| but looks obvious in retrospect when it finally clicks?
| cbm-vic-20 wrote:
| "A monad is just a monoid in the category of endofunctors,
| what's the problem?"
| Koshkin wrote:
| I don't think this definition is correct. (A monad is an
| endofunctor.)
| howling wrote:
| The definition is correct. A monad is an endofunctor with
| return and join functions. Just like a monoid in the
| category of sets is a set with identity and
| multiplication.
| crdrost wrote:
| It's correct but jargony. So in the category of sets
| there is a notion of product between two sets called the
| Cartesian product, and one can do a couple things to
| endow this product with an identity element, for example
| one might use {{}} as that object in the category of
| sets.
|
| The claim is that in other categories, there might be
| other natural combinations between two objects, for
| example a tensor product of Abelian groups combined with
| the integers Z as unit, or a composition of two
| endofunctors into a new endofunctor _F_ [?] _F_ combined
| with the identity functor.
|
| So the idea is that a monoid is somehow a destroyer of
| this combination operation; a monoid in sets un-combines
| the Cartesian product _M_ x _M_ back into the set _M_ ,
| and indeed this is a function (a set-arrow) from the
| combined objects to the underlying object.
|
| By having an endofunctor combined with a natural
| transformation from _F_ [?] _F_ back to _F_ (natural
| transformations are the arrows in the category of
| endofunctors) a monad is therefore doing exactly what a
| monoid does, if you replace the "pre-monoid" combination
| step of the Cartesian product with instead a new "pre-
| monoid" combination step of endofunctor composition.
| ianai wrote:
| LA is one of those topics that, to an extent, is built on a
| handful of core capabilities and concepts. Once you master
| those much of what follows are logical extensions or
| combinations of the previous. It goes on from there, but the
| value returned from the core material is wide reaching.
| DagAgren wrote:
| I am convinced that monads induce a very specific kind of
| brain damage that makes a person incapable of ever explaining
| monads.
| madhadron wrote:
| I've had good luck with explaining it as a characteristic
| of a programming language. In a language consisting of
| sequences of statements with bindings and function calls,
| we expect that
|
| f(x)
|
| is the same as
|
| a = x; f(a)
|
| and the same as
|
| g = f; g(x);
|
| That's the monad laws. Whatever craziness you want to put
| in the semantics, those are properties you probably would
| like to preserve in your language.
| dleslie wrote:
| Start with a container. M a
|
| Then add a way to put things in the container.
| a -> M a
|
| Then add a way to use the thing in the container.
| M a -> (a -> M b) -> M b
| jerf wrote:
| Well, you see, that's one of the problems... monad
| implementations don't have to be "containers", or at
| least not the way most people mean. This was one of the
| critical errors in many of the aforementioned
| "tutorials". IO, the quintessential monad, is not a
| container, for instance.
|
| (A nearly-exact parallel can be seen in the Iterator
| interface. You can describe it as "a thing that walks
| through a container presenting the items in order"... and
| yeah, that's the majority use case and where the idea
| came from... but it's also wrong. What it _really_ is is
| just "a thing that presents items in some order". It
| doesn't have to be from "a container". You can have an
| iterator that produces integers in order, or strings in
| lexigraphic order, or yields bytes from a socket as they
| come in, or other things that have no "container"
| anywhere to be found. If you have "from a container" in
| your mental model then those things are confusing; if you
| understand it simply as "presenting items in order" then
| having an iterator that just yields integers makes
| perfect sense. A lot of the Monad confusion comes from
| adding extra clauses to what it is. Though by no means
| all of it.)
| dleslie wrote:
| I wouldn't over-think it and over-describe it.
|
| The "aha" realization that the "container" can be an
| ephemeral concept and not resident at run time can come
| later.
|
| FWIW, I think of IO as a container: it contains the risk
| of side-effects within. All the examples you gave are
| containers in their own way.
| jerf wrote:
| The problem is telling people it's a container _is_ "over
| describing" it. We don't need to hypothesize about that.
| We have the space suits and burritos to prove it is not a
| good didactic approach. It is not removing from the
| definition to simplify, it is adding to the definition,
| exactly as I carefully showed in my description of
| "Iterator". An Iterator is "a thing that presents a
| series of items". It does not simplify the discussion of
| Iterator to say "It's a thing that presents a series of
| items out of a container, but also, it doesn't have to be
| a container". It's not the first definition that's
| "overdescribing", it's the second.
| dleslie wrote:
| Containers make sense.
|
| Abstract computer science doesn't.
|
| Part of why Haskell appears like such an implacable
| curmudgeon is the predilection of its community to
| believe that users must grasp type and logic theory to
| use it.
|
| They don't.
|
| Just like they don't need to have a mental model of their
| computer to write software for it.
| Koshkin wrote:
| In my experience, not having a mental model of the
| computer you are going to run your software on will bite
| you on the ass sooner or later.
| dleslie wrote:
| Countless mobile apps and web sites have been made with
| nigh-zero understanding of the VMs, rendering engines, or
| underlaying machine architecture.
|
| It's not the 80s anymore.
| DagAgren wrote:
| I'll just point out that neither of you have managed to
| really take a single step towards actually explaining
| monads.
| jerf wrote:
| I'm not trying, so that's not a surprise.
|
| This has inspired me to try to update my post on the idea
| in a side window, but it's been sitting on my hard drive
| for over a year now and probably still has a ways to go
| yet.
| DagAgren wrote:
| Yes, well, good example.
| vajrabum wrote:
| This explanation did it for me. https://www.reddit.com/r/ma
| th/comments/ap25mr/a_monad_is_a_m...
| jerf wrote:
| The "monad" problem is worse; many of the "tutorials" were
| actively wrong about some critical element, often more than
| one. I don't think I've seen someone claim to have linear
| algebra "click" but be fundamentally wrong about it somehow.
|
| One advantage of linear algebra is that it is, well, linear.
| Linear is nice. It means you can decompose things into their
| independent elements, and put them all together again,
| without loss. The monad interface, as simple as it is, is not
| linear; specific implementations of it can have levels of
| complexity more like a Turing machine.
| bencw wrote:
| Maybe I'm biased, but I really don't think so. Monads are
| quite a bit more abstract than the concepts in linear
| algebra. Linear algebra is both geometric and algorithmic and
| therefore very intuitive. Most of the difficulty people have
| learning linear algebra can be attributed to poor teaching
| methods.
| madhadron wrote:
| That depends on the part of linear algebra. In an abstract
| function space when you start calculating dimensions of
| kernels and the like and get ready to make the jump to
| infinite dimensions, Banach spaces, and Hilbert spaces,
| it's about as abstract as monads.
| Koshkin wrote:
| Well, to be fair, functional analysis is not part of
| linear algebra proper. (If you want to get more abstract,
| you go to rings and modules and from there to category
| theory.)
| madhadron wrote:
| That's fair. I may have a bias coming from physics
| because quantum mechanics demands Hilbert Space Now! from
| the students.
| basic_bgnr wrote:
| "whenever somebody gets a deeper understanding of monads,
| they immediately lose the ability to explain it to other" I
| don't remember where I've read this but it still holds even
| today.
| bencw wrote:
| Honestly, I think working with computers (possibly some
| programming) should be more frequently integrated into math
| courses. A computer is a natural to really interact with the
| material, like labs in the natural sciences. We're lucky to
| live in a time where this is possible, but sadly math education
| is taking it's sweet time taking advantage of this possibility.
| layoutIfNeeded wrote:
| Sounds like a case of _You Can 't Tell People Anything_:
| http://habitatchronicles.com/2004/04/you-cant-tell-people-an...
| nothis wrote:
| I guess a mathematician might look down upon sticking with 2d
| and 3d stuff because it leaves out all the interesting things
| that happen at 92382 or negative infinity. But yea, matrices
| are basically just a convenient way to write rows and rows of
| "ax + by + cz...". In linear algebra, you just do it so often,
| people made up their own syntax. And nothing can visualize it
| like transforming graphics, IMO.
|
| You don't even have to go 3D, just starting with the points of
| a rectangle in 2D and asking, "how do you put the edge points
| of this rectangle 10px to the left, rotate them 45deg and
| stretch them 200% vertically?" and you've applied a matrix.
| Even if you're not using the fancy brackets, you're using a
| matrix, and understanding it.
| dmvaldman wrote:
| I think these are good examples, but to me "linear algebra
| thinking" lies in it's generality. For example, the
| derivative is a linear operator, so how do you write it down
| as a matrix? Google's PageRank is a solution of a matrix
| equation, what does that matrix represent? Etc.
| vlasev wrote:
| > For example, the derivative is a linear operator, so how
| do you write it down as a matrix?
|
| Consider polynomials in X of degree up, but not including
| N. The powers 1,X,...,X^(n-1) form a basis. Then the
| coefficients of the polynomial can be put in a column
| vector. If D is the derivative operator, DX^n = nX^(n-1),
| so the derivative matrix can be expressed as a sparse
| matrix with D_(n,n+1) = n. Visually, it's a matrix with the
| integers 1,2,...,n-1 on the super-diagonal.
|
| You can also see that this is a nilpotent matrix for finite
| N, since repeated multiplication sends the entries further
| up into the upper right corner.
|
| You can extend this to the infinite case for formal power
| series in X, too, where you don't worry about convergence.
|
| > Google's PageRank is a solution of a matrix equation,
| what does that matrix represent?
|
| Isn't it just the adjacency matrix of a big graph?
|
| Anyway, I agree with you. Matrices and linear algebra is a
| really good inspiration for higher level concepts like
| vector spaces and Hilbert spaces and so on. That's where
| the real power lies. But even in such general domains,
| matrices are often used to do concrete computations on
| them, because we have a lot of tools for matrices.
| swiley wrote:
| I think the best thing anyone told me about linear algebra was
| that "matrices are just the coordinate form of a linear map."
| So applying the map is equivalent to multiplying it's matrix
| etc.
| Koshkin wrote:
| Reality, as always, is not that simple. Matrix analysis is a
| huge area in itself; and matrices can also be used to
| represent tensors (which generally are not seen as linear
| maps) and some other things.
| longtimegoogler wrote:
| IMO, if one is interested in a computational approach to Linear
| Algebra, Trefethens book, Numerical Linear Algebra, is the best.
|
| That book discusses the actual algorithms used for computation.
| It is a bit more advanced, but amazingly clear.
| penguin_booze wrote:
| I recently came across this rather in-depth series on linear
| algebra:
| https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS1....
| FWIW, I myself have only gone half-way thorugh part 1.
| abecode wrote:
| Two things really made linear algebra click for me: representing
| camera projections in a computer vision class and spectral graph
| theory, which basically connects graphs with linear algebra. In
| both of these, it seems like linear algebra was taken from the
| electrical engineering domain into computer science, which better
| fit my perspective.
| tomahunt wrote:
| A 2018 paper by Strang about this approach:
|
| https://www.tandfonline.com/doi/abs/10.1080/00029890.2018.14...
| vertak wrote:
| Can anyone grow why in the first 5 minutes of part 1 he shows a 3
| by 3 matrix multiples by a 1 by 3 vector yet verbally he pulls
| out of no where this idea that if you have _two_ 1 by 3 vectors
| that pass through the origin then their linear combinations can
| be represented by a plane? The jump from the 3D to the 2D has me
| lost and I gave up
| jcmoyer wrote:
| The same concept applies in 2D, which might help you build the
| intuition to understand it in 3D.
|
| If you have a vector v=(1,0) that points to the right, you can
| scale this vector infinitely in that direction by multiplying
| it by a positive scalar.
|
| 5v = (5,0)
|
| 62.1v = (62.1,0)
|
| Similarly, you can scale that vector infinitely in the opposite
| direction (i.e. left) by multiplying it by a negative scalar:
|
| -987v = (-987,0)
|
| If we call this scalar c, the expression cv allows us to
| represent any point along the X axis simply by varying c,
| meaning that cv defines a line along that axis.
|
| Similarly, we can do the same for a vector w=(0,1) along the Y
| axis, scaling it by d.
|
| Now we have a method for moving to any point on the XY plane
| simply by varying c and d in the linear combination: cv + dw,
| meaning that we've defined a plane using two vectors.
|
| Two caveats:
|
| - this won't work if v and w are parallel; for example, if v =
| -w (and neither are zero) then we can only move along a line
| instead of a plane
|
| - it also won't work if either of the vectors are zero, because
| no matter what you multiply by, a zero vector can only
| represent a single point
| swiley wrote:
| I don't know which video you're talking about but two non
| parallel vectors are enough to represent a plane, the normal
| will be their cross product.
|
| Also, if you have 3 dimensional vectors you were always in 3D.
| neutronicus wrote:
| If you multiply the 1x3 vector by all scalars from -infinity to
| infinity you get all the points on a line.
|
| If you do the same for another 1x3 vector, and it is not
| parallel to the first, you get all the points on a different
| line.
|
| These two lines define a plane (and the cross product of the
| two vectors defines its normal vector)
| sqlmonkey wrote:
| I started by only reading his book thinking it was enough. I was
| very wrong, these videos marry themselves beautifully with the
| content of the book which suddenly became incredibly clear once I
| started watching the videos. Strang teaching style can also seem
| odd at first, but don't give up, he is an amazing teacher who
| makes every concept simple to understand. This course is a true
| gift.
| chadcmulligan wrote:
| Wow, Prof Strang is 85 and still teaching! Thats very impressive
| and inspiring.
| hprotagonist wrote:
| my linear algebra professor began his first lecture with a 5
| minute rant informally titled "how you could have gotten an A in
| differential equations last semester without ever having taken
| calculus"
|
| That certainly got our attention. I've always found linear
| algebra to be kind of ... almost soothing.
| eximius wrote:
| That's a rant I'd like to hear.
| hprotagonist wrote:
| "look guys, at end of class, exam is always long list of
| silly second order system of differential equations.
|
| Well, every time, we can make so-called "guess" that solution
| looks like e*rt. Why? We know that because professors will
| only give well-behaved systems on final exams because it's
| hard to grade the other kind.
|
| So we know characteristic polynomials look like so (because
| of course they do, you can just memorize this) ... so now we
| lift out the coefficients into nifty thing called _matrix_
| and now follow these easy four steps to get roots, plug back
| in, and incidentally these are "eigenvalues", we'll talk
| about this later ...
|
| Bam. Done. A-, easy. No sweat."
| eximius wrote:
| Minus the matrix bits, that's basically how I slogged
| through DiffEq. Showing up to class was pointless because
| the prof would make up an equation to exploratively solve
| and inevitably it would be poorly behaved and the lecture
| would end with "... And and and for this kind of problem we
| have to use numerical methods".
| lcuff wrote:
| As someone who understands nothing of linear algebra, I have to
| say this "introduction" was gibberish. He may be a fantastic
| teacher, and perhaps it's a bit much to expect a 4 minute video
| to teach me anything, but it reminds me of talks from business
| people where what they're saying is obvious if you already
| understand it, and completely obscure otherwise.
| wodenokoto wrote:
| This is not a course, or a primer or an introduction on Linea
| Algebra.
|
| From the course description:
|
| > These six brief videos, recorded in 2020, contain ideas and
| suggestions from Professor Strang about the recommended order
| of topics in teaching and learning linear algebra.
| glram wrote:
| Professor Strang's lectures helped me greatly during my linear
| algebra class. I thoroughly appreciated his clear, coherent
| lecture style.
|
| On another note, he is such a nice guy. 10/10.
| [deleted]
| katzgrau wrote:
| I had very intelligent linear algebra professor in college but he
| was, in my opinion, a very poor communicator. I paid attention to
| lectures and stared at the text, but couldn't really understand
| the material. For the first part of a linear algebra course,
| students who don't mind blindly following mechanical processes
| for solving problems can do very well.
|
| Unfortunately I'm one of those people who tends to reject the
| process until I understand why it works.
|
| If it wasn't for Strang's thoughtful and sometimes even
| entertaining lectures via OCW, I probably would have failed the
| course. Instead, as the material became considerably more
| abstract and actually required understanding, I had my strongest
| exam scores. I didn't even pay attention in class. I finished
| with an A. Although my first exam was a 70/100, below the class
| average, the fact that I got an A overall suggests how poorly the
| rest of the class must have done on the latter material, where I
| felt my strongest thanks to the videos.
|
| So anyway, thank you Gilbert Strang.
| ansible wrote:
| > _I paid attention to lectures and stared at the text, but
| couldn 't really understand the material. For the first part of
| a linear algebra course, students who don't mind blindly
| following mechanical processes for solving problems can do very
| well._
|
| I had a similar, though sort of opposite experience.
|
| In high school, I breezed through the material, and started
| teaching myself calculus during the summer to prepare for
| university. Other than being a lazy student, I had no problems
| taking the 2nd semester advanced calc 2 and 3 courses my
| freshman year. I totally get what's being taught. There weren't
| a ton of practical examples, but I can easily see (for example)
| what the purpose of integration is, and how and why you'd do it
| in two or more dimensions. I could work the equations, no
| problem. Everything is great.
|
| Along comes sophomore year, and still thinking I am hot stuff,
| I take advanced linear algebra and differential equations. More
| of the same, I thought.
|
| Well... we seemed to spend the entire semester just solving
| different kinds of equations. No explanations given as to what
| they are for, where they are used, or what the point of any of
| it was. I struggled, for the very first time.
|
| I either got a D or F for the mid-term exam, which was shocking
| to me.
|
| We had _one_ chapter where we were doing something practical.
| This is where you have a water tank, and a hole in to bottom.
| Because the pressure lessens as the tank empties, the flow rate
| is not constant. However, you can solve this via diff
| equations, and I really grokked it. I finally saw the point for
| _some_ of what we had been doing. But it was just that one
| chapter, we skipped any other practical aspects for what we
| were studying.
|
| I did end up pulling out a 'C' with that class, to my relief.
| Sure, most of the blame for my lousy performance must rest with
| me, because of my poor study habits. And a little blame can go
| to the TA, who wasn't a good communicator, so that hour every
| week was kind of useless. But I also blame the material and how
| it was presented.
| hhmc wrote:
| It's probably an oversimplification, but differential
| equations -- as a field of study -- tends to be much more a
| grab bag of tricks than many branches of mathematics.
| crispyambulance wrote:
| I think that whether or not students do well, there's a
| common theme in university math curricula for non-math
| majors. Basically, math gets taught as a kind of "toolbox" of
| techniques. Unless there's a strong follow-up in subject
| matter courses (for example in engineering coursework), those
| math skills effectively evaporate.
|
| Some places use a rigorous "proof-theoretic" approach in math
| curricula. It's much harder and takes more time, but it's
| better than merely grinding on hundreds of easy
| calc-101/diff-eq problems, because students gain an
| understanding that doesn't erode as easily once they forget
| "the tricks".
|
| More CS, engineering and science students, IMHO, should
| dabble in math department courses beyond the the usual
| "required" sequence for their majors. It can be eye-opening
| and provide long lasting benefit to take a hardcore real-
| analysis course, abstract algebra or a number of other
| courses in math.
| dunefox wrote:
| > More CS, engineering and science students, IMHO, should
| dabble in math department courses beyond the the usual
| "required" sequence for their majors
|
| That was absolutely not allowed at my faculty (admittely
| computational linguistics, but I would have massively
| benefited from math courses). No courses other than the
| predefined ones, no matter how relevant. Now I have to
| learn so much afterwards, it's not even funny.
| sandyarmstrong wrote:
| Oh man, Differential Equations. After doing well in Calc 1-3
| I thought it would be no big deal. I paid attention in class
| and barely did the homework because it all seemed so
| straightforward but it was boring and I was not engaged.
|
| I came in for the first exam, sat there for maybe 15 minutes
| reading the questions, and realized I had no idea how to
| solve any of them.
|
| Luckily it was before the drop date! That was a turning point
| where I decided to only take classes that seemed fun. For me
| that was discrete math, number theory, abstract algebra, etc.
| rcthompson wrote:
| I had a very similar situation in my linear algebra course: in
| hindsight, I would literally have been better off teaching
| myself the material than listening to the professor. To this
| day it's still the main weak spot in my math/stats knowledge
| base. I'm really interested to check out these lectures.
| metreo wrote:
| Intermediate Stats writing out Chi-Squares by hand on exams
| literally ended my academic inclinations. I had been using
| software to do this for a while and something about the
| process of being forced spend hours memorizing how to by hand
| just to "earn" a letter rubbed me the wrong way. I absolutely
| know much more about Chi-squares then I'd ever need to,
| possibly an imprinting of the bad experience.
| cat199 wrote:
| haven't watched but based on summary these seem more about
| pedagogy than the subject - that said there is a full course
| worth of videos taught by same professor that are pretty good
| qorrect wrote:
| Oh yeah Gilbert Strang's original course is amazing ,
| https://ocw.mit.edu/courses/mathematics/18-06-linear-
| algebra...
| sh-run wrote:
| I took linear algebra through a community college and had one
| of those rare, really awesome CC instructors. He had spent most
| of his career at Cray and later Raytheon and then semi-retired
| as a community college instructor. He took time to make really
| great interactive Jupyter notebooks. That combined with 3 brown
| 1 blue videos really made linear algebra click for me.
|
| My only regret is that I took the class as a six week short
| course. I think my recall would be better if I had taken the
| full semester. We covered all the material, but missed out on
| the longer spaced repetition. Linear Algebra was by far my
| favorite pure math course, I hope to revisit it soon. Maybe
| Strang's lectures are the way to do that.
| exabyte wrote:
| There is a linear algebra series on Udemy called "Complete
| Linear Algebra: : theory and implementation" by Mike Cohen
| that I really enjoyed doing because he walks you through
| Matlab demonstrations (code included for Matlab and python. I
| adapt to Julia using the PyPlot wrapper for Julia).
|
| I particularly like his videos because he breaks them down
| into small bites that are easy to work into your day and he's
| a great teacher.
|
| https://www.udemy.com/share/101XOWAkYTd19WTQ==/
| synaesthesisx wrote:
| Linear algebra was one of those classes I was forced to take in
| undergrad as an engineering requirement - only to end up
| appreciating it immensely later on when I realized how many real
| world problems can be converted to matrix operations.
| anandrm wrote:
| Just curious .. what really are the usecases where of Linear
| Algebra is applied ? Any domain of software development ?
| vlasev wrote:
| It's applied pretty much everywhere. Most numerical problems
| have some linear algebra component to them. Physics uses it a
| lot too. A lot of non-linear problems have a linearization on
| which you can use linear algebra to obtain approximations.
| Ideas from linear algebra are used a lot in things like signal
| processing, quantum mechanics, etc.
| ktta wrote:
| Any fields that have anything to do with video, image, audio,
| games, machine learning.
|
| Just to have a taste of use cases: compression, filters(image
| filters for de-noising, HP & LP filters for audio),
| encoding/decoding, computer vision techniques, cryptography,
| neural nets, computer graphics (this is where most people learn
| how to use it in real computer programs)
| justinmeiners wrote:
| - scale or rotate an image.
|
| - root finding algorithm with more than one variable.
|
| - graph problems like Google's PageRank
|
| - statistical analysis
|
| - 3d rendering (projecting a 3d scene onto a 2d image)
|
| - solving systems of equation (also see linear programming)
|
| Linear algebra is very basic and fundamental to physics and
| math.
| enitihas wrote:
| Another good Linear Algebra book is "Linear Algebra Done Right",
| which Springer is giving for free right now.
|
| Link: https://link.springer.com/book/10.1007/978-3-319-11080-6
| mseri wrote:
| Came here to say that. It is a wonderful book, and I think it
| provides a more "modern" approach than the one presented in the
| videos.
| AlanYx wrote:
| Thanks for this -- do you know if there's a consolidated list
| anywhere of other books Springer is making available for free
| right now?
| enitihas wrote:
| https://link.springer.com/search/page/1?facet-
| discipline=%22...
| tvb12 wrote:
| Woah. That's a lot of books. Between the math, physics, and
| cs sections, years from now you'd look back and wonder if
| you really should have downloaded all of them.
| clarry wrote:
| There's also a free book "Linear Algebra Done Wrong," which
| might be worth checking out.
|
| https://www.math.brown.edu/~treil/papers/LADW/LADW.html
| threatofrain wrote:
| IMO Axler's book should be read either during or after you take
| an introductory course on Linear Algebra.
|
| > 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.
| vlasev wrote:
| I whole-heartedly agree. Axler's book is a great stepping
| stone to more abstract linear algebra.
| bencw wrote:
| This book is great and very much complementary to Strang's
| approach in that it leans more towards "abstract" linear
| algebra.
| brmgb wrote:
| After watching this and having read the comments, I am quite
| puzzled by the approach American seem to take to linear algebra.
| Are matrices viewed as the core of the subject in the USA ?
|
| My country curriculum introduces linear algebra through group
| theory and vector spaces. Matrices come later.
| chubot wrote:
| Yeah I would call this the engineering approach (matrices) vs
| the mathematical approach (algebra).
|
| I took like 3-4 courses in the US involving the engineering
| approach, starting in high school and continuing through the
| college as a CS major. That was all that was required.
|
| But I also like algebra, so I happened to take a 400-level
| course that only math majors take my senior of college. And
| then I got the group theory / vector space view on it. I don't
| think 95% of CS majors got that.
|
| I don't think one is better than the other, but they should
| have tried to balance it out more. It helps to understand both
| viewpoints. (If you haven't seen the latter, then picture a
| 300-page text on linear algebra that doesn't mention matrices
| at all. It's all linear transformations and spaces.)
|
| What country were you taught in? Wild guess: France?
| ojnabieoot wrote:
| I would not describe his approach as "the US approach" but it
| is a pretty standard approach to introducing linear algebra _to
| engineers,_ which is the theme of the course.
|
| I was also taught linear algebra this way, by an applied
| mathematician with a background in chemical engineering:
|
| - start by solving Ax=b with row reduction
|
| - develop theorems about linear independence and spanning sets
| of vectors based on these exercises
|
| - introduce the determinant from the perspective of linear
| systems (rather than eg geometry or group theory)
|
| - eigenvectors and eigenvalues
|
| Later I switched from physics to math and TAed a more
| "algebraic" approach involving groups/rings/fields. But the
| matrix-first approach was more helpful for both my physics
| coursework and later courses in numerical linear algebra.
| tadhgds wrote:
| I can't say whether or not it is the standard approach but I do
| know that it is very common in many countries to teach a linear
| algebra course that is heavy on matrix operations, that you can
| come away believing that linear algebra is somehow _about_
| matrices and their operations. I know many in my university
| class seemed to believe that.
|
| A book I enjoyed is Axler's Linear Algebra Done Right[0], in
| which, if I remember correctly, doesn't contain a single
| matrix.
|
| [0]https://zhangyk8.github.io/teaching/file_spring2018/linear_a
| ...
| andy_wrote wrote:
| I've recently started going through Axler carefully and doing
| the problems, a quarantine activity I guess, and have been
| enjoying it. I actually learned about this book on an older
| HN post.
|
| It does have plenty of matrices. The main thing it really
| does is avoid determinants until the very end. The
| determinant is certainly something I remember learning as a
| kind of rote operation, without really understanding any
| intuition behind why you'd multiply and add these numbers in
| this particular way. I still feel lacking in "feel" here,
| which is why I suppose I'm going through Axler now.
| threatofrain wrote:
| IMO Axler's book should be read either during or after you
| take an introductory course on Linear Algebra.
|
| > 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.
| impendia wrote:
| Yeah, math professor here, this drove me crazy.
|
| For example, I remember looking at the linear algebra book my
| department had used previously. Early on, it introduced the
| concept of the _transpose_ of a matrix:
|
| https://en.wikipedia.org/wiki/Transpose
|
| Superficially, it looks like something good to introduce. It
| is fodder for easy homework exercises, and there is a
| satisfyingly long list of formal properties satisfied.
|
| But _why_? What does the transpose _mean_? For what sort of
| problem would you want to compute it?
|
| There are good answers to these questions (see the "transpose
| of a linear map" section of the Wikipedia article I linked),
| but they are not easy for a beginner to the subject to
| appreciate.
| dntbnmpls wrote:
| > After watching this and having read the comment, I am quite
| puzzled by the approach American seems to take to linear
| algebra. Are matrices viewed as the core of the subject in the
| USA ?
|
| The US is a very big place. I doubt there is an american
| approach to linear algebra. We really don't have a single
| approach to anything. Different schools and majors probably
| approach the topic differently. My college had a linear algebra
| course specifically crafted for CS majors and engineers. I took
| that and it did focus on matrices. It was also the only math
| class that required programming. I believe math majors had
| their own linear algebra course.
|
| > My country curriculum introduces linear algebra through group
| theory and vector spaces. Matrices come later.
|
| Different strokes for different folks. If it worked out for you
| that's all that matters.
| Myrmornis wrote:
| > Different strokes for different folks. If it worked out for
| you that's all that matters.
|
| That sort of relativism is trite, specious, and insincere. If
| you don't think it matters then why are you participating in
| the discussion?
| einpoklum wrote:
| There's nothing particularly linear about groups.
| brmgb wrote:
| No but the curriculum goes from groups to fields and from
| fields to vector spaces.
| eximius wrote:
| American here. We started with the group theory and vector
| space approach, though the group theory was fairly limited to
| just enough for vector spaces as there was a separate set of
| algebra classes.
|
| It's not universal.
| currymj wrote:
| In general math departments in the US are less frightened of
| accidentally teaching the students something useful.
| praptak wrote:
| I'm currently trying to grok the finite element method. Gilbert
| Strang's explanation of the transition from the Galerkin method
| to FEM did more for me in terms of connecting the dots than
| anything else I could find on the web. And it wasn't even a
| lecture, just a kind of an interview. I think it's this one:
| youtube.com/watch?v=WwgrAH-IMOk
| kragen wrote:
| https://www.youtube.com/watch?v=WwgrAH-IMOk
|
| I feel like I don't really understand his explanation, because
| it's kind of vague. But I think that might be because you've
| seen the equations dozens of times, and I haven't seen them at
| all, so you were prepared to understand the video.
| praptak wrote:
| This makes sense. As said, it was about connecting the dots
| for me. Also, I don't even claim I fully understand FEM (or
| even Galerkin), it's just my hobby project.
| kragen wrote:
| That sounds interesting! What are you doing with it?
| praptak wrote:
| I just want to understand the magic behind static stress
| analysis. More generally I'm interested in emulating
| physics behind the rigid body model.
|
| Maybe I will create a game prototype based on the
| mechanics but this is just a vague idea.
| jp0d wrote:
| I've been doing the Statistics Micromasters from MIT. It's
| rigorous and very deep. I look forward to doing this.
| knzhou wrote:
| Gilbert Strang's linear algebra course blew my mind back in high
| school, and I still use insights from it every day. Strang has a
| particular lecturing style where he approaches every topic
| several times, often beginning many lectures before the main
| treatment. At first I thought it was a bit confusing, but later I
| realized it helped build fluency, just like a language class.
|
| I'm really thankful to MIT OCW for putting his lectures out for
| free -- in fact, I think I'll go donate to them now.
| tomjakubowski wrote:
| +1. Since I took the OCW course, whenever I multiply a matrix
| and a vector or two matrices by hand, I hear his voice saying,
| "combinations of columns." Strang must have said those words
| hundreds of times in it. His lectures stick like nothing else.
| raz32dust wrote:
| +1, I was very grateful to MIT OCW because when I learned
| Linear algebra, I could not have afforded it. Later when I got
| a job, I donated to OCW and I bought his book full price from
| his own site [1] just as a tribute to the guy.
|
| [1] https://math.mit.edu/~gs/linearalgebra/
| qorrect wrote:
| Hey me too! ( All of it )
| willbw wrote:
| Of course it is up to you what you do with your money but, they
| have a $17.5B endowment so there may be more needy causes if
| you were so inclined.
| CameronNemo wrote:
| Is there a better way to incentivize educational institutions
| to offer free content?
| bluquark wrote:
| A modest proposal: ban donating to them so they need to
| radically increase their student body (online or offline)
| to earn their keep with tuition, instead of relying on the
| donations of the extremely rich parents of legacy
| admissions students.
| michaelcampbell wrote:
| I'm in an online Master's program now. With way more
| students than I feel they can handle. I'm sure they're
| milking the tuition just fine, but when projects and
| papers aren't graded in time to determine if you should
| withdraw or soldier on, it's less awesome.
|
| While your view makes sense in the theoretical, once
| again human failings cause it to not work well in
| reality.
| blain_the_train wrote:
| What problem do you think drastically increasing the
| student body will solve?
| john4532452 wrote:
| Malcom Gladwell made a similar point on his podcast
| http://revisionisthistory.com/episodes/06-my-little-
| hundred-...
| arcturus17 wrote:
| OCW opens up top-notch education to anyone and everyone,
| regardless of social or economic background. I wouldn't take
| it for granted even with MIT's eye-watering endowment, and I
| doubt donations to it are going to be paying cafeteria
| lunches for the students.
|
| I hope you're donating and actively contributing to many non-
| profit projects and that your comment comes from being tired
| of the world's injustices rather than from callous
| impertinence, although I suspect it does not.
| Judgmentality wrote:
| I think his point is there are plenty of other noble causes
| that could use the money a lot more.
| wegs wrote:
| Most money which comes into MIT passes through overhead.
| That means if a foundation donates to MIT, a bit over 1/3
| of that money might ends up with whatever they donated to.
| A bit under 2/3 might go into the general budget (overheads
| vary by funding source, but the numbers above are from one
| specific project).
|
| On paper, overhead is used for costs of running the place.
| In practice, it's used for things like upscale faculty
| clubs, million-dollar executive salaries, $200 million
| buildings, etc. MIT has among the highest overheads in the
| academy. Ironically, MIT claims its ocean yacht makes money
| rather than losing money (which could very well be true).
|
| If you're okay with the majority of your money going to
| graft, donate to MIT. With a project like OCW, which has
| such a huge cost:benefit ratio, accepting the graft with
| the donation may be a rational decision, if you subscribe
| to a system of ethics like utilitarianism.
|
| Personally, I almost never donate to a charity where the
| highest-earner makes more than I do. I think if everyone
| did that, MIT might lose some of the graft and corruption
| which has built up there over the years.
| mattkrause wrote:
| MIT's current overhead rate is 50.5%, but that's pretty
| standard.
|
| Here's a scatterplot showing lots of research
| institutions' rates. When this was published, MIT's rate
| was slightly higher (54%).
| https://www.nature.com/news/indirect-costs-keeping-the-
| light... showing actual and calculated rates.
|
| This also applies to federal _research grants_ and is
| meant to cover costs associated with actually hosting the
| research (rent, utilities, support staff). Foundations
| can (and often do) negotiate lower rates. I'm not sure
| how donations are handled, but I don't think the same F
| &A rates apply.
| scared2 wrote:
| In High school?
| gowld wrote:
| Yes. Linear Algebra is an extension of what is commonly
| called Algebra 2 or Precalculus in high school.
|
| LA and Calculus can be studied independently in any order and
| then fruitfully combined later.
| bencw wrote:
| This is very well put. Knowledge has a hierarchical (or perhaps
| even cyclical!) structure and it's unrealistic to think that a
| body of knowledge can be taught or learned sequentially.
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