[HN Gopher] An Intuitive Guide to Linear Algebra
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An Intuitive Guide to Linear Algebra
Author : cassepipe
Score : 151 points
Date : 2022-03-31 12:03 UTC (10 hours ago)
(HTM) web link (betterexplained.com)
(TXT) w3m dump (betterexplained.com)
| galaxyLogic wrote:
| I've seen some spreadsheets, but it never occurred to me that
| they were doing matrix multiplication. Is that what spreadsheets
| typically do?
|
| For instance, in Excel is there a function for multiplying
| matrices? And getting the eigen-vector?
| chas wrote:
| Excel has the built in MMULT function, but I'm not aware of any
| built-in support for eigenvalues or eigenvectors. Many people
| have written such functions though.
|
| That said, I would be surprised if Excel spreadsheets were
| implemented as matrices. Since you can update one cell and have
| it automatically update any computation that uses that cell, I
| would expect spreadsheets to be implemented with some sort of
| dependency graph so it's easy to traverse and update the values
| that need to be changed. (This could be implemented as an
| adjacency matrix, but I haven't seen that representation used
| before for programming language dataflow analysis.)
| galaxyLogic wrote:
| Interesting idea. I wonder would it be possible to create a
| spreadsheet-app where every "sheet" was a matrix and then you
| would apply linear algebra operations between the sheets to
| produce output-sheets. Would that be all a spreadsheet-app
| needs? It might be a simpler unifying design for spreadsheet
| programming.
|
| The problem I've had with Excel etc. is that every cell can
| hide an operation and some do and some don't and it becomes
| difficult to understand what the totality of calculation is.
| Whereas if it could be expressed as operations between
| matrices the whole calculation could be expressed as a single
| formula, perhaps. (?)
| laerus wrote:
| the book is totally worth it, thanks to Kalid Azad for making
| these concepts easier to understand.
| kuharich wrote:
| Past comments: https://news.ycombinator.com/item?id=4633662
| nh23423fefe wrote:
| These kinds of explanations are so meh to me. Linear algebra is
| useful once you begin to look for vector spaces you didn't know
| you had.
|
| Thinking of matrices as spreadsheets is barely abstraction.
| Seeing the derivative operator represented as a matrix, acting
| over the polynomial vector space can open your eyes.
|
| Taking the determinant of that matrix shows that d/dx isn't
| invertible.
|
| Thinking of the fixed point of the transformation yields exp, the
| eigenfunction of the operator.
| pfortuny wrote:
| Yep.
|
| I used to teach this. One of the key ideas is to get rid of 3d
| geometry and state, from the beginning, huge sized problems
| (simple models of traffic using kirchoff's laws, image
| convolution, statics...). Otherwise, why define the
| determinant? Just compute it. Or eigenvalues? Or kernels?
| lookingforsome wrote:
| I read it as a more intuitive primer on the subject, which I
| think it did fairly well imho.
| snapetom wrote:
| Concepts you described can't happen without a Step 1 in even
| approaching linear algebra. These types of explanations help
| many take that first step.
| actually_a_dog wrote:
| Right, and that's a perspective you pick up on in a second
| course in linear algebra, typically. The key insight really
| is that the core concept is that of a vector _space_ , rather
| than vectors _per se_. The only thing we really ask of
| vectors is that it be possible to apply linear functions with
| coefficients from your favorite field to them. Other than
| that, vectors themselves aren 't that interesting: it's more
| about functions to and from vector spaces, whether it's a
| linear function V -> V or a morphism V -> W between two
| different vector spaces.
|
| This is actually a common theme of mathematics, that the
| individual objects are in some sense less interesting than
| maps between them. And, of course, the idea that any time you
| have a bunch of individual mathematical objects of the same
| type, mathematicians are going to group them together and
| call it a "space" of some kind.
|
| In fact, my previous paragraph is pretty much the basis for
| category theory. One almost never looks at individual members
| of a category other than a few, selected special objects like
| initial and terminal objects. A lot of algebra works in a
| similar way. If I could impart one important insight from all
| the mathematics I've read, done, and seen, it would be this
| idea of relations being more important than the things
| themselves.
| imachine1980_ wrote:
| You need mathematics abstract to understand this and when you
| have it you already know most of this.
| tawaypol wrote:
| I don't know about you but Linear Algebra was the first
| abstract mathematics I was ever exposed to.
| lookingforsome wrote:
| Right, my exact sentiment.
| whimsicalism wrote:
| I honestly don't see anything about this website that is
| really building intuition.
|
| The "derivative operator" notion that the GP is describing
| was hugely important for me in intuiting what linalg could
| do.
| qorrect wrote:
| > The "derivative operator" notion that the GP is
| describing was hugely important for me in intuiting what
| linalg could do.
|
| Do you have a link someone could read more about this ?
| voldacar wrote:
| Here is a nice short video on how that works:
| youtube.com/watch?v=2iK3Hw2o_uo
| u_y wrote:
| Spot on.
|
| If I may add, I found "useful magic" like discrete Fourier
| transforms, local linear approximations and homogenous
| differential equations as exciting examples to motivate
| students into the abstract theory of linear transformations
| capn_duck wrote:
| Yes I think this spreadsheet view is so detrimental and
| confusing for newcomers. I'm not even sure the analogy makes
| sense. The key part of linear algebra imo is the concept of
| linear transformations.
|
| T(a+b)=T(a)+T(b)
|
| Matrices just happen to be one way of expressing those
| transformations.
| chas wrote:
| And for extra magic, since every vector space has a basis,
| every linear transform between vector spaces with a finite
| basis can be represented by a finite matrix
| (https://en.m.wikipedia.org/wiki/Transformation_matrix).
| While this might feel obvious if you haven't explored
| structure-preserving transforms between other types of
| algebraic objects (e.g. groups, rings), it is in fact very
| special. Learning this made me a lot more interested in
| linear algebra. It unifies the algebraic viewpoint that
| emphasizes things like the superposition property (T(x+y) =
| T(x) + T(y) and T(ax) = aT(x)) with the computational
| viewpoint that emphasizes calculations using matrices.
|
| Since all linear transforms between vector spaces with a
| finite basis are finite matrices, the computational tools
| make it tractable to calculate properties of vector spaces
| that aren't even decidable for e.g. groups. For a simple, but
| remarkable example: All finite vector spaces of the same
| dimension are isomorphic, but in general, it's undecidable to
| compute if two finitely-presented groups are isomorphic.
| lookingforsome wrote:
| I really enjoyed this, almost read as a primer in less academic
| order of operations and something more natural in the form of
| intuitive learning. Thanks for sharing!
| melling wrote:
| Any thoughts on acquiring the skills needed to understand linear
| algebra so it's possible to read Axler's Linear Algebra Done
| Right
|
| https://linear.axler.net/
|
| ... or Mathematics for Machine Learning
|
| https://mml-book.github.io/
|
| There are YouTube videos for both books:
|
| Axler:
| https://youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmSinFV...
|
| MML:
| https://youtube.com/playlist?list=PLiiljHvN6z1_o1ztXTKWPrShr...
| plandis wrote:
| Perhaps not what you're looking for but if you can get through
| Griffith's Quantum Mechanics you likely can get through Axler.
| I found it helpful to draw examples from QM when self studying
| in Linear Algebra Done Right.
| peterhalburt33 wrote:
| Huh, somehow I also learned LA best through Griffiths QM. I
| almost wish Griffiths would write a Linear Algebra textbook.
| the__alchemist wrote:
| This seems like a way of viewing a small subset of linear algebra
| (matrix multiplication). My favorite approach is 3 Blue1Brown's
| visual one, also avail on Khan Academy.
|
| This article leaves out the key insight of matrices as a
| transformation of space.
| peterhalburt33 wrote:
| I would also add that one fundamental aspect of linear algebra
| (that no one ever taught me in a class) is that non-linear
| problems are almost never analytically solvable (e.g. e^x= y is
| easily solved through logarithms, but even solving xe^x=y
| requires Lambert's W function iirc). Almost all interesting real
| world problems are non-linear to some extent, therefore, linear
| algebra is really the only tool we have to make progress on many
| difficult problems (e.g. through linear approximation and then
| applying techniques of linear algebra to solve the linear
| problem).
| whimsicalism wrote:
| This is why physics should be taught along with math.
|
| They went out of their way to explain how first-order linearity
| was so fundamentally important for all sorts of non-linear
| forces.
| lordleft wrote:
| An amazing blog that has made a lot of math more accessible to
| me.
| triyambakam wrote:
| A prominent sentiment in the comments here is that this resource
| isn't that good. I only studied up to Calculus II, so what would
| be a good resource to approach LA?
| peterhalburt33 wrote:
| It really depends on what you'd like to learn LA for, and how
| comfortable you are with abstraction: LA can span from concrete
| multiplication of matrices and vectors all the way to very
| abstract (e.g. vector spaces over general fields or even
| modules over rings). I know many people recommend Gilbert
| Strang's introductory linear algebra (I have not read it, but
| it seems to fall into the former camp), but I might also
| recommend Sheldon Axlers Linear Algebra Done Right. In all
| honesty, I learned La the best from David Griffiths quantum
| mechanics text, although it is not a comprehensive in its
| coverage of the subject (not that it should be, given that it
| is a physics text). I guess I am trying to say that there are
| many different flavors and interpretations of linear algebra,
| and while matrices and vectors may be simple at first, it does
| tend to rob the subject of its richness and depth (e.g. what do
| these matrices represent, what are the canonical structures
| etc.) so I am a bit biased towards going full generality at
| first, and perhaps reading a more rote computation book on the
| side (I understand we all have limited time though).
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