# Determine all vector subspaces of the real vector space $\mathbb{R}^2$

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 $\operatorname{span}\left\{\begin{pmatrix} 1\\0 \end{pmatrix}, \begin{pmatrix} 0\\1 \end{pmatrix}\right\}$?

• Yes, but maybe it is asking for a more geometrical answer... Commented Apr 21, 2013 at 23:07
• Example answer: The $x$-axis is a subspace of the plane. This is just $\text{span}\{(1,0)\}$. Commented Apr 21, 2013 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? Commented Apr 22, 2013 at 0:55
• For 1 dimension subspaces, it's either span(1,0) or span(0,1)? Is that right? Commented Apr 22, 2013 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?
– user70962
Commented Apr 22, 2013 at 21:49

## 1 Answer

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?