Let $V$ be an $n$-dimensional vector space over $\mathbb R$. Prove that $V$ has a subspace of dimension $r$ for each $0 \le r \le n$.
Is this as simple as saying that $V$ has a basis of $n$ elements, and then take $r$ elements from this basis and this will generate an $r$-dimensional subspace? Or am I missing something? It seems too simple.
Thanks
Best Answer
For sake of having an answer, yes, if $\{v_1,v_2,\ldots,v_n\}$ is a basis of $V$, then $\operatorname{span}\{v_1,v_2,\ldots,v_r\}$ is an $r$-dimensional subspace of $V$ for every $0\le r\le n$ (with $\operatorname{span}\emptyset=\{0\}$ by convention).