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If i have two vectoes [1,2] and [3,7] what will be the column space ? Will it be a plane or a line ? In the case of [1,2] and [3,6] it is the line y=2x . Can the column space for 2-d vectors be a plane ? If so in what conditions ?

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What do you mean by column space? –  Davide Giraudo Feb 1 '12 at 10:00

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First of all, a note on terminology: as far as I know, the term column space is usually reserved for matrices, i.e. we talk about the column space of an $m \times n$-matrix $A$, which is the subspace of $\mathbb{R}^n$ spanned by the column vectors of $A$. I'll assume that what you're after is the subspace of $\mathbb{R}^2$ spanned by the vectors $[1,2]$ and $[3,7]$, more precisely a basis for it (if not, please edit your question to make it clearer).

Also, this looks like a typical homework question for a linear algebra course, so I'll only give a few hints: since you mention column spaces, you've probably covered column spaces of matrices, and also how to find a basis for this space (and hence dimension). If so, you can solve the first part of the problem by first finding a matrix whose column space is the space you are trying to determine, and then find find the column space of this matrix.

To gain some more intuition, and to answer the other questions, here are some more hints: what is the dimension of a plane? What about a line? And what is the definition of the dimension of a vector space? Does the set $\{ [1,2], [3,7] \}$ constitute a basis for a subspace of $\mathbb{R}^2$ (in fact, it would have to be a basis for $\mathbb{R}^2$ itself)? How about $\{ [1,2], [3,6] \}$? It is probably helpful to draw a few pictures...

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