Taking Linear Algebra this semester. When will I be enlightened?

>>8454574
eigenvectors and eigenvalues

>>8454574
Matrix Algebra or Vector Space Theory?

>>8454574
when you realize, that matrices are fucked up, if they map more/any other than the zero vector onto the zero vector, because you can't invert such a matrix (their linearity forbids to map a zero vector to a non-zero vector). therefore you need to check, if your matrix is fucked up, and that's why there exist such things like a determinants

>>8454597
your "fucked up" is just the usual case, it's invertible matrices that are special. and it's not at all surprising that many linear maps have nontrivial kernel: think of any projection R^3 -> R^2; by rank arguments that one can't be injective.

>>8454625
thanks for adding this. this leads to the following true statement: most matrices are fucked up

>>8454634
QED

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>>8454574
>When will I be enlightened?
When you read this.

when you realize linearization can solve so many problems in STEM that you might as well just apply those matrix algorithms all day long and do nothing else

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never

>>8454673
got a link ?

>>8454625
>>8454634
This actually isn't true. "Most" linear maps are invertible. The determinant is a polynomial in the entries of the matrix, and the linear map is invertible if and only if the determinant is nonzero. So the only time a matrix isn't invertible is the special case that the determinant is zero.

The fact that the determinant is a polynomial makes this argument work. If you think of the space of matrices as a really large Euclidean space, then the equation "determinant = 0" is like a "single restriction" or the "removal of a free variable." You end up with a "1-dimensional" thing while the invertible matrices make up a "n^2-dimensional" thing.

I'm assuming we're talking about square matrices here. If you're going to talk about other matrices then we don't have a notion of a determinant.

>>8455284
Sorry, the invertible matrices make up a "(n^2 - 1)-dimensional" thing where the matrix is n by n.

>>8455284
Also, non-square matrices are never invertible. So if you are just looking at all matrices then yeah most are fucked up. But not really, "most" non-square matrices still have full rank, which is the best one can hope for.

>>8455177
http://gen.lib.rus.ec/search.php?req=katznelson+linear

>>8454574
I don't understand the /sci/ meme about Linear Algebra being some kind of ultimate math-pill.

Can someone explain it to me? Isn't it just about solving linear systems of equations?

>>8455725

At a low level, yes. From what I understand, mathematicians enjoy the relative power linear algebra offers in regards to viewing such problems abstractly: namely, linear algebra affords the opportunity to better abstractly model problems in other areas of mathematics.

I'm not a pure mathematician (I study computer science), so I only appreciate linear algebra in as much as it applies to my specific domains (cryptography, machine learning, and complexity).

>>8455733
As a CS undergrad: how is linear algebra related to complexity?

>>8454574

Elementary linear algebra isn't the good stuff, it's just a routinezed formalization of assorted basic notions, a necessary organizational process for people in the sciences to be aware of.

In an undergraduate curriculum, multivariable calculus, say, is literally more mathematically interesting than the notions in linear algebra. But the value of learning linear algebra is that the learning process drills-in and generalizes in ways that are applicable for absolutely everybody (programming, spreadsheets, moving data around).

You're grinding through some intermediate mathematics, anon. no worries.

>>8455704
>Yonantan Katznelson
hey that guy lectures at my university. he likes very corny jokes. funny to sew him here on /sci/

Probably quaternions and geometric transformations.

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>>8454574
yea but in the next course
the most practical and useful class on can take is numerical linear algebra

>>8455284
>>8455286
THIS IS SO NOT TRUE. adding one constraint such as "det = 0" frequently raises the COdimension by one, making the set of non-invertible matrices into a (n^2-1)-dimensional thing. this is not the case however, because 0 is not a regular value of the determinant function (its differential isn't always surjective), so the dimension can actually differ in different points. Whether the set of singular matrices even has a well defined dimension is beyond the scope of my current knowledge.

it's true that "most" square matrices are invertible. this is best expressed by saying that GL(n) is an open dense subset in the set of all invertible n by n matrices, therefore it's an n^2-dimensional object.

>>8456224
***GL(n) is an open dense subset in the set of ALL n by n matrices.

>>8455725
I dunno, here in germany Linear Algebra is the first math class students have to take and it's guaranteed to weed out many freshmen due to being so different to the school maths they are used to. It's not especially hard or anything though.

>>8454668
Kek