# what is the column space for 2-d vectors

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

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).
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...