From this question:
Suppose $V$ is a vector space with dimension $6$. Let A and B be subspaces of V with dimensions 4 and 5 respectively. What are the possible values for the dimension of A intersection B?
The reasoning given in the answer:
The best way is to look at the basis. Any basis for $A$ consists of $4$ elements, say $\{a, b, c, d\}$. Suppose that $e, f$ are vectors of $V$ such that $\{a, b, c, d, e, f\}$ span the whole $V$. Since $B$ has dim $5$, exactly five elements of $\{a, b, c, d, e, f\}$ are in $B$.
$\ldots$
The part in bold bothers me. Why is it necessary that any basis for $B$ must contain elements from basis of $V$ ?
More generally, what is the relation between basis vectors of a vector space to those of its subspace?