Proof for basis and subspace

Question

Prove or disprove the following statement:

"If $(v_1,v_2,v_3,v_4)$ is a basis for the vector space ${\Bbb R}^4$ and $\textbf{W}$ is a subspace of ${\Bbb R}^4$ then some subset of $(v_1,v_2,v_3,v_4)$ forms a basis for $\textbf{W}$."

I have an intuition for this and I think it is false because it's saying that I can write $(v_1,v_2,v_3,v_4)$ as a linear combination of each other since the vectors form a basis. However, just because $\textbf{W}$ is in he subspace of ${\Bbb R}^4$ does not mean I can write is with linear combinations of $(v_1,v_2,v_3,v_4)$. There is no guarantee that I can completely span $\textbf{W}$ with $(v_1,v_2,v_3,v_4)$ right?

Is that even the right approach or is my thinking completely off? How would I actually do this mathematically?

Answer 1

Your idea is right. A concrete counter example would be good.

Let $v_i = e_i$, the $i$-th standard unit vector.

Let $W = \operatorname{span}\{ v_1 + v_2\}$, then none of the $\{e_i\}$ is a basis for $W$.

Answer 2

Actually, since the set $\{v_1,v_2,v_3,v_4\}$ has $2^4=16$ subsets, there are only $16$ subspaces of $\mathbb{R}^4$ which have a basis which is a subset of $\{v_1,v_2,v_3,v_4\}$. Since $\mathbb{R}^4$ has infinitely many subspaces…