I'm studying hyperplanes in my Linear Programming course.
Knowing the properties of a hyperplane, suddenly came to me the hypothesis that a hyperplane is an affine space or a vector space (but we can consider an affine space a generalized version of a vector space).
Is this true?
By definition, given $A$ affine space of dimension $n$, its hyperplane is an affine subspace of dimension $n-1$.First of all note that every $K$-vector space, given the homomorphism: $f\colon V\times V\to V$ for whitch $f(v,w)=w-v$ determinates an affine space structure on V (in other words you can think every vector space as an affine space with the origin in $(0,0,......0)$ and that particular affine structure I just described. In less rigorous terms think to affine spaces as traslations of a determinate vector space: evrey finitely generted vector space is spanned by vectors belonging to a basis of the space itself. So, using the same basis as directional vectors and moving the origin to a definite point of A( a set of points) you get an affine space. Note that if the traslation=id (identity function) then i'm not moving the vector space from its origin, but still it can be considered as an affine space as I explained. Hope this is clear. For any doubts, just ask!