THE BORE
General => The Superdeep Borehole => Topic started by: recursivelyenumerable on December 19, 2011, 01:48:50 AM
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I admit it, I kinda suck at linear algebra, or just basic matrix operations for that matter. I can never remember which is rows and which is columns and blah blah blah. So I'm going to abuse this message board to ramble to myself about matrices for a while.
OK. Go go go go go go go!
So an m X n matrix means m ROWS and n COLUMNS, right? And such a matrix represents a linear transformation between n-element vectors to m-element vectors, i.e. m X n matrix = n -> m transformation. So you need to mentally reverse the order of the indices ... which is confusing. But then if you put the n-element vector to the LEFT of the matrix and do matrix multiplication, thinking of it as a 1 X n matrix, the matrix does then act as an n -> m linear transformation ... but unfortunately, that goes against the usual convention where the operation goes to the left of the thing it's operating on (e.g. f(x), or even in normal algebraic notation for multiplication like 2x, the style kind of emphasizes the '2 ...' as an operation on x even though it could go either way).
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I did some math in matrixes once a long long time ago. Was definitely fun in a puzzle solving playing with numbers way.
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then, I guess the uhhhh monoid? (matrices, matrix multiplication) is uhhh homomorphic? to the monoid (linear transformations, composition). But again, what are the appropriate row/column orders and left/right ... nesses ... here?
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So if we take an n -> m linear transformation, that corresponds to an n X m matrix put to the right of the vector it's transforming or an m X n matrix put to the left of the vector, we can compose it with an m -> uhhhh p-for-some-p transformation, to get an n -> p xformtion.
I guess
xAB : x in R^n, A is n X m, B is m X p, AB is n X p, xAB in R^p
or
BAx : x in R^n, A is m X n, B is p X m, BA is p X n, BAx again in R^p
ok, I think I get it. sort of. I'm sure this will last all of an hour or so and then I'll get confused again
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now to try and understand conjugate gradient (http://en.wikipedia.org/wiki/Conjugate_gradient_method).
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To this day I think the spectral radius (http://en.wikipedia.org/wiki/Spectral_radius) has the coolest name of any mathematical concept I've studied.
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ooh, eigenvalues
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So were you working with infinite-dimensional vector spaces, Pringo?
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So were you working with infinite-dimensional vector spaces, Pringo?
iirc they were finite dimensional but it's been a while now so I can't say for sure.
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you know, somehow eigenvalues make more sense to me when I think about them in terms of general abstract vector spaces than when I think of them in terms of general vectors in Euclidean space or something, guess I'm weird like that :lol
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Let off some steam, Bennett.
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sleep is for the little people
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Don't remember much of matrix multiplications (Only that the transpose and determinant can only be defined for square matrices :P). I remember loving linear algebra though, best part of the maths I took fo sho. recursive :bow2
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Thought it said MAURICES.
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just "lost it" AGAIN and trying to regain it by reading this thread. apparently I am just keeping my notes on evilbore from now on, hope demi maintains the archives & search feature well
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Thought it said MAURICES.
each time I see this thread I think this :'(
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I took Linear Algebra. Like Differential Equations, I got nearly a perfect score in the class because you have to realize that none of it makes any sense. The more you try to make sense of it, the more you get tangled up.
I still have no idea what eigenvalues and eigenvectors are.
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the problem seems to be with my understanding of matlab syntax rather than my understanding of matrices themselves.
linear algebra and differential equations make NOTHING BUT sense. YOU don't make sense. unfortunately matlab doesn't make sense either :(
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the problem seems to be with my understanding of matlab syntax rather than my understanding of matrices themselves.
linear algebra and differential equations make NOTHING BUT sense. YOU don't make sense. unfortunately matlab doesn't make sense either :(
I haven't worked in MATLAB in forever but isn't everything there defined to work with matrices and vectors by default? I remember the first time I used it it took me a while to realize how to use scalar values.
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kinda, I think the problem really is that I just don't understand the scoping and precedence rules.
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yeah, of all math, linear algebra and diff calc have the most practical application and are the most sensible. on the other hand, laplacian transforms and e-fields :(
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just "lost it" AGAIN and trying to regain it by reading this thread. apparently I am just keeping my notes on evilbore from now on, hope demi maintains the archives & search feature well
I am pretty confident that this is the only thread in the history of this message forum to have the word "Matrices"
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It classes up the joint
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Never took Linear Algebra in school, so I was thrown in on the deep end when I had to convert a bunch of MATLAB routines to C++... but it wasn't too bad in the end. Just had to read some guides to do things.
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http://www.youtube.com/watch?v=IRsPheErBj8
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ok, it's an identity that (A' * B')' = B * A, right? I think? or something like that? and I understood WHY at one time, but now I will have to re-derive it AGAIN. but hey, the common trigonometric identities eventually sunk in for good after I re-explained them to myself for the 389th time, maybe these will too?
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so A: Am -> An (YES I AM USING POSTFIX FOR FUNCTION APPLICATION)
B: Bm -> Bn
ugh not sure this will work though, I think "the transpose of a matrix" doesn't mean much to me other than a mechanical operation and that might be the real problem
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maybe I should try thinking of 2x2 in geometric terms
like
x->x x->y
y->x y->y
x->x y->x
x->y y->y (transpose)
soooooooooooo transpose is about reversing the uhh "direction of contribution"?
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still h8 matlab, but I have started resorting to using HUNGARIAN NOTATION to keep track of what is a column vector and what is a row vector, etc. Works surprisingly well in maintaining my sanity.
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wait you're having problems with matlab being column major?
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I don't know what that means. Because Matlab is dynamically typed I tend to lose track of what orientation everything has, whether they're scalars or vectors or matrices, whether I need to transpose, etc.
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Works surprisingly well in maintaining my sanity.
We'll be the judge of that, sir.
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i want to reply to this thread, but like so many computing related threads it makes me think of work and i go "ehhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh"