# Subspaces of an $\Bbb R$-vectorspace

What is a proper subspace of an $n$-dimensional $\Bbb R$-vectorspace $V$? It is a subset of $V$ that is a vectorspace in its own right correct? How would we denote these vectorspaces, surely as a subset they aren't $\Bbb R$-vectorspaces of their own.

Could we take the $\Bbb Q$-vectorspace as a subspace of $V$?

• When you talk about subspaces the scalar field doesn't change. If $V$ is a vectorspace over $\mathbb{R}$ then all subsets of $V$ that are also vector spaces over $\mathbb{R}$ are subspaces of $V$.
– R_D
Oct 3 '15 at 12:09

Consider the case when $n=1$. $\mathbb{R}$ is an $\mathbb{R}$ - vector space. The only vector subspaces of it are $\{0\}$ and $\mathbb{R}$ itself. $\mathbb{Q}$ is a subset of $\mathbb{R}$ and is a $\mathbb{Q}$ - vector space but is not an $\mathbb{R}$ - vector space.
You are right when you say subspaces of an $n$ - dimensional $\mathbb{R}$ - vector space $V$ are subsets of $V$ that are vector spaces in their own right. However when you say it is a vector space in its own right the scalar field is always fixed. So a better way to say it is that subspaces of $V$ are subsets of $V$ that are $\mathbb{R}$ - vector spaces in their own right.
Secondly, proper subspaces of $V$ are proper subsets that are also $\mathbb{R}$ - vector spaces. So a proper subspace of $V$ is a $k$ - dimensional subspace of $V$ where $k < n$. (If $k = n$ then the subspace is no longer proper). How would you visualize such a space? Well if you think of $V$ ar $\mathbb{R}^n$ then the $0$ - dimensional subspace is just $\{0\}$, the $1$ - dimensional subspaces are lines through the origin, $2$ - dimensional are planes through the origin and so on (This is also mentioned in another answer)
Indeed a proper subset of V that is a $\mathbb{R}$ vector space is a proper subspace and the scalar field is fixed. Note in $\mathbb{R}^2$ {0}, and the lines passing through the origin are the only subspaces of $\mathbb{R}^2$. Similarly in $\mathbb{R}^n$ the only subspaces are {0}, the lines passing through the origin, planes containing the origin, the n-k (k=1,...,n-3) hyperplanes containing the origin.