# is it possible to categorize all subspaces of a given vector space?

For example, if my big space is $P_3(\mathbb{R})$, what are all the possible subspace of this vector space?

Obviously $P_3(\mathbb{R})$ itself, $P_2(\mathbb{R})$, $P_1(\mathbb{R})$ and $\{0\}$ are subspaces, but then what if I choose something random like:

$\{x^3-2abx + abc : a, b, c \in \mathbb{R}\}$

or

$\{ax^3-2b-c+(2a-b) : a, b, c \in \mathbb {R}\}$

They might or might not be subspaces, so we check. But can we compile a list of all possible forms a subspace might take?

• Apr 12 '17 at 23:00
• The key word you want is Grassmanian en.wikipedia.org/wiki/Grassmannian Apr 12 '17 at 23:01
• Is it just the fact that the vectors are polynomials that is throwing you off; do you have an easier time wrapping your head around, say, all subspaces of $\Bbb R^4$? Apr 13 '17 at 3:05
• I suppose it was, but I realized they're the same. That is, $P_3(\mathbb{R})$ is isomorphic to $\mathbb{R}^4$. Apr 13 '17 at 3:07

Theorem: Let $F$ be a field and let $k$ be a positive integer. Then every $k$-dimensional vector space over $F$ is isomorphic to $F^k$.
It's not particularly hard to prove that $F^k$ has a subspace of dimension $F^\ell$ for all $0\leq\ell \leq k$ by looking at coordinate projections onto a basis. Thus we know that the subspaces of $F^k$ are precisely $\{F^\ell:0\leq \ell\leq k\}$. Counting the number of each subspace is also possible.