←back to thread

The Little Book of Linear Algebra

(github.com)
468 points scapbi | 2 comments | 02 Sep 25 14:17 UTC | HN request time: 0.001s | source
Show context
andrewla ◴[02 Sep 25 18:02 UTC] No.45106778[source]▶
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 matrices and Gaussian elimination, all the while bored out of your skull, until you have the tools to really start to approach the interesting areas.

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 until one day when you're looking at a chain of linear transformations you realize that everything clicks.

This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains.

It's funny because it's just a pedagogical nightmare to get students to care about any of this until one day two years later it all just makes sense.

replies(20): >>45106971 #>>45106998 #>>45107045 #>>45107321 #>>45107368 #>>45107522 #>>45107638 #>>45107686 #>>45108661 #>>45108864 #>>45109019 #>>45109881 #>>45111057 #>>45112025 #>>45112656 #>>45112834 #>>45113034 #>>45113260 #>>45114881 #>>45126195 #
jcranmer ◴[02 Sep 25 20:35 UTC] No.45108661[source]▶
> 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 as a description of how to convert one sense of basis vectors to another set of basis vectors, and for that, you first need to understand how vectors are described in terms of basis vectors.

replies(1): >>45109621 #
nh23423fefe ◴[02 Sep 25 21:55 UTC] No.45109621[source]▶
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.

replies(2): >>45110169 #>>45110434 #
1. jcranmer ◴[02 Sep 25 22:54 UTC] No.45110169[source]▶
If I write a vector v = [1, 3, 2], what I am actually saying is that v is equal to 1 * e₁ + 3 * e₂ + 2 * e₃ for three vectors I have previously decided on ahead of time that form an orthonormal basis of the corresponding vector space.

If I write a matrix, say, this:

  [[1  2]
   [3  4]
   [5  6]]
What I am doing is describing is a transformation of one vector space into another, by describing how the basis vectors of the first vector space are represented as a linear combination of the basis vectors of the second vector space. Of course, the transformed vectors may not necessarily be a basis of the latter vector space.

> The natural motivation of matrices is as representing systems of equations.

That is very useful for only very few things about matrices, primarily Gaussian elimination and related topics. Matrix multiplication--which is what the original poster was talking about, after all--is something that doesn't make sense if you're only looking at it as a system of equations; you have to understand a matrix as a linear transformation to have it make sense, and that generally means you have to start talking about vector spaces.

replies(1): >>45118112 #
2. nh23423fefe ◴[03 Sep 25 17:03 UTC] No.45118112[source]▶
Doesn't make sense is too strong though.

If you have a system Ax=y and a system By=z there exists a system (BA)x=z

This system BA is naturally seen as the composition of both systems of equations

And the multiplication rule expresses the way to construct the new systems' coefficients over x constrained by z.

The C_i equation has coefficients which are the evaluations of the B_i equation over the A_k-th coefficients

C_ik = B_ij A_jk

concretely

        A11 x1 + A12 x2 = y1
        A21 x1 + A22 x2 = y2

        and

        B11 y1 + B12 y2 = z1
        B21 y1 + B22 y2 = z2

        then

        B11 (A11 x1 + A12 x2) + B12 (A21 x1 + A22 x2) = z1
        B21 (A11 x1 + A12 x2) + B22 (A21 x1 + A22 x2) = z2

        rearrange and collect terms

        (B11 A11 + B12 A21) x1 + (B11 A12 + B12 A22) x2 = z1
        (B21 A11 + B22 A21) x1 + (B21 A12 + B22 A22) x2 = z2
the coefficients express the dot product rule directly