Vector Spaces and their Subspaces are studied in Linear Algebra, while Subsets are a much more general formulation (subspaces are subsets of appropriate vector spaces with certain properties). You visualize these things concepts by doing lots of problems on them.
Thanks, I've read those over a couple times before and I get the math of it. But, let's say and egg.
Are any of these statements true about the egg?
-The eggs as a whole is a Vector Space and the yolk sack is a Subspace?
-If there was something else within the yolk, we could say that the yolk sack is a Vector Space and the thing inside is a Subspace.
Sorry to use an egg as an example, it just seems to fit.
In Euclidean space, Rn, with addition and scalar multiplication defined "componentwise" ((x, y, z)+ (u, v, w)= (x+u,y+ v, z+ w) and r(x, y, z)= (rx, ry, rz)) vector spaces are flat. The sphere, \(\displaystyle x^2+ y^2+ z^2= R^2\), for example, is not a vector space because (1, 0, 0) is in that set but 2(1,0,0)= (2,0,0) is not.
The only subspaces of R3 are planes containing (0,0,0) and lines containing (0,0,0).
(Lines and planes that do NOT contain (0,0,0) are often referred to as "linear manifolds".)