I want to open this question with a disclaimer, that I am not a native english speaker and I did not study linear algebra in english, and am simply trying to use words that I believe best translate from my own language.
Today in our online class we were discussing the orientation of real non-infinite vector space V, and it having 2 directions as every hyperplane in it has "two-sides" as there is no continuous way to get from one side to the other, I am imagining this hyperplane as the space over and below a simple plane (like a sheet of paper).
This direction is defined by taking Alpha basis of the space prescribing it the positive direction and then finding a basis Beta whose transsition matrix would have a negative determinant.
Finally my question. Let's say we have a vector in the hyperplane that is going in the positive (up) direction, we can find a vector that goes in the same but negative direction. That is when the V is "two-side". (like poking a stick through the paper, the part of the stick over the plane is the positive and the part of the stick below the paper is the negative direction vector)
What happens, when the space is one-sided like a mobius loop? When I look at a classic mobius loop (I imagine it like the strip of paper that is glued back twisted), and have a hyperplane over it, I can also find a vector that goes in a positive (up) direction and find a vector that goes opposite to it. I understand that the stick is now poking from the same "side" but the individual parts are still going in the opposite direction, no? Is mobius loop space even "pokable"?
Please explain to me like I am 5, this quarantine is not doing wonders for my ability to ask dumb questions.
The Mobius strip $M$ is a topological manifold, roughly meaning that it has a vector space of tangent vectors attached to each point (and this attachment is continuous in a sense). In the case of the Mobius strip, these are 2 dimensional spaces: the tangent planes of the surface.
Now if we pick an appropriate basis $e_1,e_2$ in the tangent space of a point, within this tangent space we can't move them continuously to $e_1,-e_2$ because that has different orientation.
However, if we continuously move $e_1,e_2$ through the tangent spaces along a circle of the strip, we'll arrive to $e_1,-e_2$.
When such phenomenon is present, the manifold is considered not orientable, as it doesn't have a simultaneous orientations given at once on each point which are locally compatible with each other.