# Visualizing Subspaces and Subsets (in R3)

I have trouble visualizing what exactly these are. Vector Space, Subset, Sub Space...
What's the difference and how can I "see" it. I'm a very visual person.

Related Linear and Abstract Algebra News on Phys.org
http://en.wikipedia.org/wiki/Vector_space#Definition"

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.

http://home.scarlet.be/~ping1339/Pvect.htm
http://www.cs.odu.edu/~toida/nerzic/content/set/basics.html
http://library.thinkquest.org/C0126820/setsubset.html

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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.
Thanks!

HallsofIvy
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.