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I'm being asked to determine "all vector subspaces of the $\mathbb{R}$ vector space $\mathbb{R}^2$". What does that even mean? Isn't it simply the $\operatorname{span}\left\{\begin{pmatrix} 1\\0 \end{pmatrix}, \begin{pmatrix} 0\\1 \end{pmatrix}\right\}$?

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Yes, but maybe it is asking for a more geometrical answer... – Marra Apr 21 '13 at 23:07
Example answer: The $x$-axis is a subspace of the plane. This is just $\text{span}\{(1,0)\}$. – noobProgrammer Apr 21 '13 at 23:27
The 2 dimensional subspace and 0 dimension subspace are trivial. How would you describe the set of 1 dimension subspaces? What would a basis of such a subspace look like? – Tpofofn Apr 22 '13 at 0:55
For 1 dimension subspaces, it's either span(1,0) or span(0,1)? Is that right? – mercurial Apr 22 '13 at 18:55
@mercurial span$\{(1,0)\}$ will only be the $x$-axis which is a subspace. How about span$\{(1,1)\}$? Is that a subspace? A line through the origin will be a subspace. How can you describe all lines that go through the origin? – BU982T Apr 22 '13 at 21:49
up vote 0 down vote accepted

The span you give is $\mathbb{R}^2$ itself. It is a subspace but you're missing others. By the dimension theorem, there exists subspaces of $\mathbb{R}^2$ of dimension 0, 1 & 2. Does that help?

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