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Essence of linear algebra [video] (youtube.com)
140 points by adamnemecek on Nov 27, 2016 | hide | past | web | favorite | 26 comments



Hopefully some friendly admin can come along and fix the link.

The rest of the videos from 3Blue1Brown are pretty awesome as well: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw


I keep hearing about "linear algebra" and although I'm a third year engineering student and am finished with all of my dedicated math courses, I've never taken a course with that title. Looking at this guys videos, it seems like it's just "matrices and vectors" which I've done over the years during high school and in my Calculus courses.

It that all it is, or am I misunderstanding. Everybody seems to think that this field of math is very important, I feel like I should know it by now.


It can be tempting to reduce linear algebra to 'matrices and vectors', but there is more to it. Often times you can find a 'matrix algebra' course or something along those lines (I remember taking 'computational matrix algebra'), which really does pretty much just discuss computations with matrices and vectors. Linear algebra has more to do with the properties of vector spaces and 'linear transformations' on vector spaces—which can be represented using matrices, but there are deeper underlying concepts than the matrix computations themselves.


It is a very practical field of math, especially for computer scientists (Graphics, some variants of machine learning, pretty much any large scale computation with chunks that can be processed in parallel), which is probably where people derive importance from. Another huge benefit is that linear algebra (For matrices with certain properties, not generally) often allow for iterative and approximate solutions, which can make linear algebra a more tractable computational route. Some of your engineering classes may have used numerical solvers that rely on linear algebra to iteratively obtain relatively precise solutions to difficult equations.

You are correct that the fundamentals (Vectorized computation, aka matrices) are often introduced early on. The level at which you interface with linear algebra often scales with the level of math you are at. You can add them [1], integrate them [2], and use them to solve some PDEs [3]. You can see how all these "levels" of linear algebra would be accompanied by another math class pushing you to utilize linear algebra at a higher level than before. It appears that you're at the first level. I've actually never done [2] in the classroom, but [3] was introduced to me in a class on differential equations that utilized matrices as a possible solution route.

If you are interested in learning more, Khan academy has good linear algebra videos.

[1] http://www.purplemath.com/modules/mtrxadd.htm [2] https://www.youtube.com/watch?v=z73ed-bd9ek [3] http://www.maths.manchester.ac.uk/our-research/research-grou...


There are many excellent courses available online if you wished to take a look.

See MIT OCW Scholar's Linear Algebra offering: https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algeb...

Take a look at the final exam and see if it seems like something you've seen. If not, I strongly urge you to consider studying the subject.

A book that shows linear algebra's applications at a more advanced level is Strang's Computational Science and Engineering: http://math.mit.edu/~gs/cse/


>, I've never taken a course with that title.

I don't know about UK curriculum but the colleges in USA like MIT/Stanford/Princeton/etc have courses focusing on linear algebra (matrix math). Example:

https://explorecourses.stanford.edu/search?view=catalog&filt...

One of those classes is often taken as an elective to meet one of the math requirements for graduation.



Delete the last character, in the link. The question mark.


Seems like youtube isn't HTTP URL standard compliant.


I noticed that linear algebra courses are typically offered in colleges after calculus. Is there a reason why?


You need knowledge of calculus sooner in a typical undergrad curriculum for most majors. For example, freshman physics assumes you're taking Calc I/II concurrently. Most college kids can't handle two math classes at a time, so they take linear algebra after.


In addition to the other reasons already mentioned, many believe that calculus further develops the ambiguous quality known as mathematical maturity so students who take calculus before linear algebra will be better prepared for the more rigorous course than those who jumped in.

(Here I use "calculus" to refer calculus courses without an emphasis on proofs.)


in my experience calculus is a good bridge between your average math-background in school and linear algebra. In some sense calculus is the natural extension of school math (at least in germany), its more complicated, more rigorous and more abstract, but still familiar. For example you still want to differentiate/integrate, but maybe in R^n. You also touch a bit of the most fundamental topics of linear algebra in calc 2 when you transition from R to R^n and (in my experience) develop some fundamental understanding that help you with linear algebra.

My roommate is studying math and music to be a teacher and i advised him to take calc first and then linear algebra and until now it's working out.

Source: Currently studying CS Bsc. in Karlsruhe where we are required to take Calc 1 and 2 and Linear Algebra 1 and 2, but in parallel (so first Calc 1 and LA 1 and then Calc 2 and LA 2, it's also doable).


Perhaps its tough for the average student?!


I failed to see the point of linear algebra when I studied in during my CPGE but it only took a couple of sessions of a Numerical Analysis class a year after to make me look past the tedious abstractions and realize its true beauty. (Compression (SVD), LU decomposition etc)


Can anybody recommend a linear algebra course based in finite fields?

A lot of theorems like SVD and eigenvectors seem like they'd carry over, but they are always proven over complex numbers.


I think you want a text like Hoffman-Kunze which gives the basic definitions and theorems over an arbitrary ground field. At some point the field starts to matter -- giving the real plane a quarter turn has no real eigenvalues, for example; and I don't see a good way to transport the statement of SVD to the setting of finite fields.


The first third of Harvard's Math 23a includes finite fields in Linear Algebra. It is available as an online class. https://www.extension.harvard.edu/academics/courses/linear-a...

For example, see https://youtu.be/c194lY79jOM minute 16:30

The course makes no specifics for finite fields but does illustrate that linear algebra works for any field, finite or not.


I believe one of the first theorems you go about proving in linear algebra is that given a vector space of a given dimension you can find a function that preserves the linear structure to any other vector space of the same dimension. After proving that they just proceed to work with complex numbers since every other (finite dimensional) vector space is the same in that sense.


One possible approach is to take a class/read a book on coding theory. Algebraic block codes rely heavily on linear algebra over finite fields.


That is probably a good idea, but I'll always be wondering about the linear algebra tools they didn't use. Were they just not applicable? Or do they not work over finite fields?



broken?


It is. But it has 4 votes :)



Thanks! We've updated the link.




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