Matrices

This may be a brainlet question, but what are matrices? And why is their multiplication defined the way it is?

I ask because I'm an engineering student and we use matrices for pretty much everything, but the only understanding I have is that they are an easier way to express systems of equations.

What other meaningful things can be said about matrices that would help me understand why we use them so much? I'm looking for "intuitive" (or dumbed down) explanation, avoiding big words if possible (like, "they are a bijective homotomorphic function that blah blah...)

Other urls found in this thread:

youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/
mathpages.com/home/kmath638/kmath638.htm
twitter.com/SFWRedditVideos

youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

Here you go, my son.

Well, in my view, the easiest way to put it is that they are a bijective homotomorphic function that blah blah...

thank you for that, i already watched that playlist. It's clearly very well made and I understood a few things, but I still feel that I'm missing the whole point, to be honest

read up the basics of shader coding to get an idea how it can be useful

ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

it's like 25 1-hour lectures, but the Lecturer is really good, and there's a whole lot to know about matrices.

An n x m-matrix is just a linear function from some m dimensional space to some n dimensional space.

Because the function is linear, you can break it it into linear component functions. For example, imagine some [math]f: \mathbb{R}^2 \to \mathbb{R}^2, f(x,y) = (2x,3x+y)[/math].
Now, you can break the above linear function into two component functions, also linear, like so,
[math]f_x: \mathbb{R}^2 \to \mathbb{R}^2, f(x,y) = (2x,3x)[/math] and [math]f_y: \mathbb{R}^2 \to \mathbb{R}^2, f(x,y) = (0,y)[/math], where the sum [math]f_x +
f_y = f[/math]. (The sum of two linear functions is always a linear function).

What a matrix basically is, is one such [math]f[/math] broken into a component form, where each column represents a component function. The first column would represent the function [math]f_x[/math] and so on.

Thus, the matrix for our function [math]f[/math] from above would be [math]\begin{bmatrix} 2 & 0 \\ 3 & 1\end{bmatrix}[/math]

So what's the deal with matrix multiplication? It's actually just the composition function of two linear functions.
[math]\begin{bmatrix} a & c \\ b & d\end{bmatrix}\begin{bmatrix} o & q \\ p & r\end{bmatrix} = \begin{bmatrix} oa+pc & qa+rc \\ ob+pd & qb+rd\end{bmatrix}[/math]
This is how it would look like "normally".
[math]f(x,y) = (ax+cy,bx+dy)[/math] and [math] g(x,y) = (ox+qy,px+ry)[/math]
Stick [math]g[/math] in [math]f[/math] and the following happens.
[math]f(g(x,y)) = f(ox+qy,px+ry) = (a(ox+qy) + c(px+ry),b(ox+qy) + d(px+ry))[/math]
Which simplifies to [math]x(oa+pc,ob+pd) + y(qa+rc,qb+rd)[/math]
If you were to write that in a matrix form, you'd get the same matrix we got from the matrix multiplication earlier.

That's how the formula for matrix multiplication is derived and what matrix multiplication actually means. We use matrix multiplication because it's more efficient and matrices have some interesting properties that would be more difficult to see from just the components of some linear function.

Get better intuition on vector spaces, do proofs and you will never ask this again. We have this textbook on linear algebra in my language, written by a string theorist, it's somewhat difficult but i think it emphasises the right parts to give you great intuition. It's unlike any other textbook where the intuition had to be found by me, here it radiates to you. The only other book that does the same is Shilov's lingebra, Hatcher's algebraic topology and Landau's theoretical physics bible.

Alright this is the good shit, thanks. I'm gonna read your post a few more times but I think I understand what you're saying. Thanks!

a linear map is the nicest map possible, examples include rotation, reflection, scaling and shearing - pic related. as user have said, matrices are precisely linear maps and the matrix multiplication is composition of the corresponding maps. this makes those maps very easy to understand and work with. whenever something rotates in a video game, it means that coordinates of that object are being multiplied by some matrix.