I am a bit confused, since I passed my course in linear algebra few years from now.
If I can make a vector space over field R which has angles as elements, then angles are vectors.
So i know that angles can't be vector, but I see vector space (V, +, *), where set V contains some angles in Radian or whatever and I can see that these condition work.
1) (V,+) is a group
2) a*(u+v) = a*u + a*v
...
and so on by definition.
So what am I doing wrong?
Let's also make that V goes [0,2pi) to make unique neutral.
Yes. And?
but they shouldn't be, right? intuitively
>>7712963
Anything that follows vector space laws is a vector.
You might think of it as the space of rotations in R^n.
>>7712963
The way you define a vector in something like physics is nothing close to the way you define them in pure mathematics.
>>7712978
Even in physics you constantly deal with "non-intutitive" vectors like in Hilbertspaces or spherical harmonics.
>>7712920
what. the. actual. fuck.
vector space : field acting on an abelian group if you want to be pedantic. Down to earth, V is a k-vector space whenever u+v and au are in V for any u,v in V and a in k. There are a fucking shit load of vector space where elements are not "vector" in the usual sense (function spaces, distribution spaces,...). The "field" in the definition is not an optional detail. Generalization of this are called module over a ring.
Angles : plane angles form an additive abelian group. Definition involves equivalence classes of couple of unit plane vectors. You can build an isomorphism with the group SO(2) or U and then to R/2piZ to define a good notion of measure for angles.
Where do they form a vector space ?
>>7712977
What's with that we had in physics classes: "Vector has to have magnitude and direction"? Angle doesn't have direction, or can I say that it's direction is based on it's magnitude?
>>7712984
Ok, dude. in mathematical sense, angle in plane is a vector since it is a element of vector space, function f is vector in C(R) (for example). I get that. But where do you make difference between "intuitive" vector and element of vector space vector?
Im trying to explain it to my dad, he is physicist.
>>7712988
Well, strictly speaking vectors are things that are defined by the vector laws and "arrows" with length in direction are just the most prominent examples of vector spaces.
Maybe you can think of an abstract vector space as something that sort of behaves like arrows (linearity, basis etc.).
Direction in this sense just means that a vector consists of different basis "entries". In R^3 these are multiples of the "unit directions" x,y,z. In other spaces they might look completely different.
Btw I didn't check whether angles really are vector spaces.
That's always problem with algebra. It has same fucking nouns for all the shit it creates
>>7712984
>Where do they form a vector space ?
Isn't the vector space of angles isomorphic to the vectors on a unit sphere (where the addition is defined by "mod 2pi" or something?
>>7712920
As long as it satisfied the axioms of a vector space, you're golden.
>>7712988
>What's with that we had in physics classes: "Vector has to have magnitude and direction"?
Strictly speaking it's incorrect. However the notion of angle and hence some sense of "direction" is inherited from a suitably defined inner product on your vector space as such: cos(t)=<f,g>/(<f,f><g,g>) for vectors f,g and t in [0,pi). The inner product also induces a norm, ||f||=sqrt(<f,f>) describing magnitude.)
>>7712993
>make difference between "intuitive" vector and element of vector space vector
The way kids are introduced to vectors are as points in real 2 or 3 space, obviously equipped with a norm (pythagorean theorem) and the "dot product." General vector spaces are just a little more abstract, since they may not have suitable norms or inner products. The parallelogram and stretching/shrinking rules kinda go out the window if you don't define length, but if you can conceptualize a topological basis generating a topological space, you shouldn't have much trouble with vector bases generating vector spaces. It's almost the same idea - you constructively describe all of the sets elements
>>7712988
No, you can have a vector space on [math]\mathbb{R}[/math] for fuck's sake.
You can literally have a vector space on the set [math]\{0\}[/math]
>>7713183
Or so I thought, but I checked, and I was mistaken.
this entire thread lmao
>>7712920
>Vector
>a quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.
Vectors have angles so maybe something having to do with that?
>>7713396
In a quadratic, two dimensional plane, vectors only exist relative to an origin in space meaning that there's an inherent angle relative to the two dimensional space. Whereas if you're dealing with one dimension then there can be no angle.
In topological space there's always an angle of a vector pertinent to the origin.
>>7712988
>Angle doesn't have direction
>>7712977
rotation composition is not commutative tho. Vector addition is
>>7713396
It would be equially weird as if trying to make vector space (R,+) over field X. what X would you take
>>7713396
>Vectors have angles
>>7712920
The problem is you are confusing a Geometric vector with all other vectors. In Linear algebra a vector isn't something that just "points somewhere" it's a space of objects that share the same characteristics. It's a shame that your professor didn't stress this enough, because I've seen a lot of professors trying to connect geometric vectors and vectors in Rn, but what they should be doing is treating them on their own terms.
>>7714015
I agree. At first I thought problem could also be my language, but now that you mention it, both of those structures are called vectors.
I guess what I wanted to ask in the beginning is whether angle (in HS sense) is a scalar?
>>7714015
Vectors in Rn are babby first vector
>>7712920
angles can be vectors, you just showed it yourself
>vector space (V, +, *)
> V contains some angles
>(V,+) is a group
>a*(u+v) = a*u + a*v
