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The Little Book of Linear Algebra

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Re: The Little Book of Linear Algebra

#71

Post by jdshaffer »

tptacek wrote: post #29
andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

I didn't think any part of linear algebra was boring. I was hooked from the moment I saw Ax=b => x = b/A. Gaussian elimination is a blast, like an actually-productive Sudoku puzzle, and once you have it down you can blaze through the first 2/3rds of an undergrad linear algebra course. I don't consciously try to gain automaticity with math subjects, but matrix-column multiplication I got pretty quickly and now I just…

Didn't know about Gaussian elimination until today. Really cool, and really useful! Thanks for sharing!

Re: The Little Book of Linear Algebra

#72

Post by griffzhowl »

nh23423fefe wrote: post #61
jcranmer wrote: post #53

Earlier quoted context omitted.

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. It's only difficult if you are wedded to a description of matrices and vectors as seas of numbers that you grind your way through without trying to instill a fuller understanding of what those numbers actually mean. The definition makes a lot more sense when you see a matrix a…

I dont agree with this. Matrices don't convert sets of basis vectors to sets of basis vectors. What would you say about singular matrices for example? The natural motivation of matrices is as representing systems of equations.

There's no single best way to understand any of this, but the action of a matrix on the standard basis vectors is a totally reasonable place to start because of its simplicity, and then the action on any vector can be built out of that because they're linear combinations of basis vectors.

Re: The Little Book of Linear Algebra

#73

Post by griffzhowl »

barrenko wrote: post #8

Tried to pick a book to get into linear algebra recently, the experience was fairly hellish. First course this, second course that, done right, done wrong... I'd to the LADR4e route, but I don't have the proof-it chops yet...

I like Serge Lang's books for clarity of explanations. He has an Introduction to Linear Algebra which concisely covers the basics (265 pages in the main text), and grounds the matrix computations in the geometric interpretation.

Be aware that Lang has another book, called just "Linear Algebra", which is more theoretical.

Re: The Little Book of Linear Algebra

#74

Post by sureglymop »

tenacious_tuna wrote: post #51
tptacek wrote: post #29

Earlier quoted context omitted.

I didn't think any part of linear algebra was boring. I was hooked from the moment I saw Ax=b => x = b/A. Gaussian elimination is a blast, like an actually-productive Sudoku puzzle, and once you have it down you can blaze through the first 2/3rds of an undergrad linear algebra course. I don't consciously try to gain automaticity with math subjects, but matrix-column multiplication I got pretty quickly and now I just…

> I didn't think any part of linear algebra was boring. My formal linear algebra course was boring as hell, to me. The ~4 lectures my security prof dedicated to explaining just enough to do some RSA was absolutely incredible. I would pay lots of money for a hands-on what-linalg-is-useful-for course with practical examples like that.

There's a book called ILA (Interactive Linear Algebra) that I found really good: https://textbooks.math.gatech.edu/ila/

Re: The Little Book of Linear Algebra

#75

Post by viewtransform »

cosmic_cheese wrote: post #35
srean wrote: post #17

Earlier quoted context omitted.

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away In my experience it need not be like that at all. One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these fol…

If anybody is aware of materials that teach linear algebra via graphics as suggested here, I would be interested to hear about them. As someone who learns best through practical application, maths have been by far among my greatest weak points, despite having written software for upwards of a decade. It’s limiting in some scenarios and pure imposter syndrome fuel.

3Blue1Brown [Essense of linear algebra](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...)

Re: The Little Book of Linear Algebra

#76

Post by rramadass »

barrenko wrote: post #8

Tried to pick a book to get into linear algebra recently, the experience was fairly hellish. First course this, second course that, done right, done wrong... I'd to the LADR4e route, but I don't have the proof-it chops yet...

You might want to checkout the book Practical Linear Algebra: A Geometry Toolbox by Dianne Hansford and Gerald Farin (its 1st edition was simply named The Geometry Toolbox: For Graphics and Modeling) to get an intuitive and visual introduction to Linear Algebra.

Pair it with Edgar Goodaire's Linear Algebra: Pure & Applied and you can transition nicely from intuitive geometric to pure mathematical approach. The author's writing style is quite accessible.

Add in Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares by Stephen Boyd et al. and you are golden. Free book available at https://web.stanford.edu/~boyd/vmls/

Re: The Little Book of Linear Algebra

#77

Post by lll-o-lll »

phforms wrote: post #38
WillAdams wrote: post #19

Earlier quoted context omitted.

Nice pairing the text with 3Blue1Brown's lectures on linear algebra! https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

As an autodidact who never learned this stuff at school/uni, his lectures are what made linear algebra really click for me. I can only recommend them to anyone who wants to get a visual intuition on the fundamentals of LA. What also helped me as a visual learner was to program/setup tiny experiments in Processing[1] and GeoGebra Classic[2]. - [1] https://processing.org - [2] https://www.geogebra.org/classic

Nitpick: No one is a visual learner, or more correctly everyone is. Multimodal is the way, so good teachers will express the same concepts in several ‘modes’ and that helps develop the intuition.

Re: The Little Book of Linear Algebra

#78

Post by ndriscoll »

andrewla wrote: post #18
Chinjut wrote: post #15

Earlier quoted context omitted.

Why do you say it's practically impossible to motivate matrix multiplication? The motivation is that this represents composition of linear functions, exactly as you follow up by mentioning. It's a disservice to anyone to tell them "Well, that's the way it is" instead of telling them from the start "Look, these represent linear functions. And look, this is how they compose".

Sure, that's a way to approach it. All you have to do is stay interested in "linear functions" long enough to get there. It's totally possible -- I got there, and so did many many many other people (arguably everyone who has applied mathematics to almost any problem has). But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fac…

Linear functions are the ones that we can actually wrap our heads around (maybe), and the big trick we have to understand nonlinear problems is to use calculus to be able to understand them in terms of linear ones again. Problems that can't be made linear tend to be exceptionally difficult, so basically any topic you want to learn is going to be calculus+linear algebra because everything else is too hard.

The real payoff though is after you do a deep dive and convince yourself there's plenty of theory and all of these interesting examples and then you learn about SVD or spectral theorems and that when you look at things correctly, you see they act independently in each dimension by... just multiplication by a single number. Unclear whether to be overwhelmed or underwhelmed by the revelation. Or perhaps a superposition.

Re: The Little Book of Linear Algebra

#80

Post by Sleaker »

andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

I went through Khan academy on linear algebra a long time ago because I wanted to learn how to write rendering logic, for me I was implementing things as I learned them with immediate feedback for a lot of the material. Was probably the single most useful thing I learned then.
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