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I understand how to manipulate vector spaces and subspaces and how to prove various statements about them, but I still don't fully understand what they represent geometrically.

I just need an intuitive grasp as to what these are. Is a vector space a geometric area that contains all possible vectors of the field $\Bbb F$? For example, if $V$ is a vector space over $\Bbb R^3$ then does that mean $V$ contains all vectors in three-dimensions that are part of $\Bbb R$? But then what is a subspace of $V$? Would that perhaps be a plane? Would it be a vector space in $\Bbb R^2$?

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The way I think about it is that, vector space is a collection of objects. We have defined (our own) way of "linearly Combining" given any $N$ objects and any $N$ numbers from field $F$ for every $N$. Then, if every such possible linear combination gives a object inside the set, then its a vector space. Now as for a subspace. It is a strict subset of the original set, which has the same properties as the orginal set. – dineshdileep Jan 28 '13 at 17:56
Wikipedia has a very good article on Vector Spaces. Why not have a read... – Fly by Night Jan 28 '13 at 17:56
up vote 4 down vote accepted

The classic example of a $3$-dimensional real vector space is the Euclidean space $\mathbb R^3$. Its $2$-dimensional subspaces are precisely all planes passing through the origin, and its $1$-dimensional subspaces are precisely all lines passing through the origin. The origin itself forms the singleton $\{0\}$ which is the only $0$-dimensional subspace.

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