/sci/ - Science & Math

https://www.khanacademy.org/math/precalculus/vectors-precalc Follow this link

What is a vector?

>>9826516
Without getting extremely pedantic, in a simple 2d spacial form it's just a coordinate... Really.

I can choose a point, say x=10, y=10, and draw a line from that point to the origin x=0,y=0, put a little arrow on it and it becomes a 'vector'.

The line you drew gave it a length, or magnitude, and the point x=10,y=10 itself gave it a direction. It's more neat and compact than talking about lines because a line doesn't have a length associated with it.

This gets expanded generally into more useful definitions and people realized at some point you can do lots of math and physics shit with it.

What else do you want to know?

>>9826542
Let me add to that, and this tripped me up a bit, a vector an be drawn from ANY point. I chose the origin (0,0) in that example as the point it's drawn from but when you study other stuff you realize a vector can exist at any point.

>>9826531
Elements of a vector space. :^)

>>9826516
number with a direction,
can also represent an area
https://youtu.be/f5liqUk0ZTw?t=1m12s

>>9826556
That's a perfectly valid answer, there's no need for your smug smiley.

>>9826516
>>9826528
>>9826542
>>9826561
>those brainlets that still work with geometric vectors

>>9826561
this video is fun
>need to learn about tensors, but they are confusing as fuck
>always the same exact bullshit explanation
>"you know what a vector is? well tensor is like the same thing only there's like more of them"
>stumble across this video, looks promising
>dude does a great job explaining vectors
>finally I will understand tensors
>he goes on and on about vectors, the video is near end
>last minute
>"so this is what a vector is. and tensor is like the same thing, only there's like more of them"

nigga it's just the pythagorean theorem. A squared plus B squared equals C squared, this easy shit nigga.

>>9826516
Just use matrices and you won't have to know jack shit about what's actually going on.

Start with the fact that a vector is just a 2×1 matrix denoting the change in X as the top element and the change in Y as the bottom element with magnatude being equal to length. It can be placed anywhere on a graph, but is generally drawn starting at the origin unless otherwise specified.

That should be about all you need, just make sure your matrix algebra techniques are compatible, otherwise you'll need to swap "row" for "column" and vice versa. Go nuts

>>9826516
For the applications you'll use, vectors are mostly so obvious in how they work that there's nothing to explain.
Learn about dot products and cross products, they're super-useful. I love dot products so much.

I just want someone to explain to me htf eigenvectors work

>>9826564
It's a valid answer, but it's not a good answer

>>9826966
It is a good answer, OP just asked a bad question. He should have asked what a vector space is.

a direction from a point you choose

>>9827048
Everyone knows that vectors have both direction and magnitude

>vectors
Nah, what you want to learn first is Modules

>>9826531

Does anyone else find it interesting how vectors are a somewhat new concept? They weren't unified with the rest of math until the end of the 1800s. Like how the fuck did they even do physics?

>>9828187
The same way ancient romans built arches without the use of calculus.

>>9828187
It's also interesting that they came from the imaginary part of quaternions, which are in fact bivectors!

>>9827149
if you equip your vectorspace with an innerproduct, that is

>>9826531
it's a measuremeant of a particular direction

>>9826998
kys

>>9828225
Geometry and trial and error?

Theyre just arrows lol

>>9826516
Vectors in what sense? CS? Physics? Or math?

>>9828387

no, the arrows are representants, vectors are classes that contain all representants

it's actually not simple to give a good answer, and introducing vector spaces makes a lot of sense

>>9828357
Brainlet

>>9826531
>>9826966
Wrong. It's the only way to discuss vectors.
There is no notion of a vector without the entire vector space satisfying the vector space axioms. None at all. Thus all there really *is* to discuss is the space, and calling its elements "vectors" is just a convenient manner of speaking.

>>9826516
> be me
> learning about the rigid body and that kind of things
> we saw inertial matrix in the theorical class, but we didn't really understand the tensor part
> my three professors have a PhD in physics
> "well guys this seems like a matrix but it isn't really a matrix cause it's a tensor"
> we asked her what is the difference
> "Hummmm it's quite the same but tensors have more properties"
> ...
> She asks the other professor that is also studying math if he knows how to define a tensor
> Silence
> "Well guys we aren't going to ask you how to get the inertial tensor we will just give it to you as a number so don't care about that"
> Mfw

>>9829146
>physicists trying to explain mathematical concepts
Physics professors are worse at explaining than any maths or engie professor I've ever had. Why is this?

>>9826516
>>9829103
A vector space V is a collection of objects, A, B, C,..., an operation (+) that satisfies the following properties. We take it that s,t,u,... are scalars.
1) A+B is always a member of V
2) sA is always a member of V
3) if t=1, tA=A
4) A+B=B+A
5) A+(B+C)=(A+B)+C
6) there is some member of V called 0' (not to be confused with the number 0) that has the property that B+0'=B for all members in V
7) for any vector (that's what we'll call these objects from now on) C, there is a -C in V so that C+(-C)=0'
8) s(tA)=t(sA)=(st)A
9) s(A+B)=sA+sB
10) (s+t)A=sA+tA

Hope that answers the question OP
10

>>9829146
I don't understand why physicists still teach "tensors". Geometric algebra is a rich and important subject, but the "tensor" part is mostly formalism that is ignored by most people at the end because its tedious. Tensors as a way to talk about coordinate-free stuff is really more of a historical curiosity as all the definitions can be taken as coordinate free, and it just happens a lot of objects that are used act linearly.

>>9829553
Do tensors in physics have anything to do with tensor products of Hilbert spaces/modules/etc?

>>9829572
Well, the obvious answer is in QM in which the idea of tensor product of hilbert spaces has a fundamental physical interpretation as the hilbert space of the tandem system. But "tensors" have no unique physical meaning, that's my point and in many cases the important properties are not really related to lineartiy. Obviously linearity is a good property, but the whole development of "tesors" usually is just formalization. The idea was that with coordinate dependent notation and definitions, it's not clear that certain shit is a well defined physical object as the coordinates changing changes the value, so the development of "things that transform like x" was done, but somewhere in history, they found through analysis, differential geometry/topology and abstract algebra that the objects can be defined a priori without any notion of a coordinate system. Yes, the tensor product has a pretty important part in QM, but that's a pretty specific case that is possible because the equations governing QM are linear.

>>9829553
But they actually didn't teach us tensors, they just say something like "oh well there is a tensor which is something similar to a matrix but we don't care about it, so calm down kids we will give it to you as a number" I felt like in HS.

>>9826516
an ordered set that can be combined with other ordered sets through binary operations which produces another ordered set

>>9829476
As an engineer: because most mathematical concepts take time to explain and basically meaningless to engineers, they are just tools, if you now how to use them and have a vague understanding of how they work you are good to go.
Im not saying this is a good way to teach, or learn! It is what it is.

>>9826838
Consider your favorite n-dimensional real-valued vector space, like say the Euclidian plane. There are two standard basis vectors: <1,0>, the unit x-vector, and <0,1>, the unit y-vector. A linear transformation takes a vector to another vector in a way that respects scalar multiplication and vector addition. Its action on an arbitrary vector <x,y> is determined by its action on those basis vectors, since if v is a vector and Av represents v transformed by a linear transformation A, we have

A<x,y> = A<x, 0> + A<0,y> = xA<1,0> + yA<0,1>

So we see that it is sometimes of interest to select particular vectors to understand the general action of a linear transformation. There are certain vectors which are not changed by a linear transformation, or just stretched/scaled (and not rotated). These are eigenvectors, and the amount that they are stretched by is the corresponding eigenvalue.

Eigenvalues have loads of applications. For example, a matrix is invertible if and only if 0 is not an eigenvalue. If an n-dimensional linear transformation has n distinct eigenvalues, we can say more stuff. Things like that.

>>9826723
geometric vectors are vectors, there's no better way to teach vectors than to start off with them as an example.

>>9831159
Vectors "starting" somewhere besides the origin is a concept that cannot be represented in the vector space definition