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I am confused about vector spaces and subspaces. I've just started a book on linear algebra, and i understood the 1st chapter which delt with gaussian reduction of systems of linear equations, and expressing the solution set as matricies, but the 2nd chapter deals with vectors and i'm perplexed.

I've learnt that if the solution of a system of equations has 3 unknowns, and 2 free variables for example, then the solution is a plane in 3 dimensional space. Now, is that 3 dimensional space a vector space? Is the plane a subspace? I really don't understand the concept of vector spaces... the book just lists 10 proofs that something is a vector space but i'd like to know what it is in geometric terms. Thanks.