/sci/ - Science & Math

>>12592315
BieleFELD not BieleFIELD, silly!

>>12464568
No we don't. We have it in high school.

>>12178882
I constructed a freaky chain complex and was interested in its homology. Start with a module, take its injective envelope, then project onto this injection's cokernel, then quotient out all the fixed points under a certain subgroups action on the cokernel. Then take the injective envelope of this quotient of the cokernel and define your differential to be the composite of these two projections and finally the injection. The image was easy because that is just two surjections followed by an injection, but the kernel was a bit more tricky. It is actually the sum of the kernel of the first projection and the preimage of the kernel of the second projection, call those [math]K_1, K_2[/math]. Then I wanted to consider the quotient [math](K_1 + K_2)/K_1 \cong K_2/K_1\cap K_2[/math]. There were some technicalities here, like how I wanted a certain subgroup to act trivially on this, and indeed it does. The trivial action was far from obvious in the sum form but the action is trivial on [math]K_2[/math].

>>11900786
>Well I had my shameful geometric vector questions.
The most shameful question is that one never asks but is tormented by for the rest of their life.
>They can be considered as points if their beginning is the origin (in some spaces only apparently), fine, and they are also (directed) line segments because (I guess) they have a beginning and an end point and as such you can put a line through them.
Yeah, you get the line segments with arrow tip pointing to some direction from the points and then realising that you can move either of its end points freely (as long as you do that to the other one also). This follows from the fact that the change in each coordinate is the same whether you start from the origin or some other point.
>Is what I said thus far correct?
Seems good.
>So, does a geometric vector also codify a set of points (as a line segment)?
Both (I mean they are equivalent so yeah) codify a set of points, namely a line. You can take any vector [math]v\neq 0[/math] and define [math]L_v = \{ rv \ |\ r\in\mathbb{R}\} [/math] to get a line going to the direction of the given vector through the origin. If you know the length of your vector, you can use that and this set to obtain an explicit set of a points on a line segment starting at the origin and ending as far from it as you want. To move it around, you just do the translation like you had the lines y = kx + c back in school to move it around in your Euclidean space. A thing worth noting though: a translation is not a linear map unless it is the identity!
>Which then somehow are equivalent to other geo vectors with the same lenght and direction?
Yes. The point is that your geometric vector anywhere is determined by the change in each coordinate, and this change is independent of the points themselves.

I hope I managed to ask your questions somewhat sufficiently. I am assuming we are talking about the Euclidean spaces. I hope the English isn't too broken.