Sorry, I might be asking a stupid question, but somehow I can't convince myself of the assertion in the title:
Let $\{u_1,\ldots,u_n\}$ be a basis for a vector space $(V,F)$ and suppose $S$ is an $m$-dimensional subspace of $V$. Now, if the first $m$ vectors in the basis happen to be elements of $S$, can we conclude that they form a basis for $S$?
My guess is a yes, and I would justify it by showing these $m$ vectors are linearly independent in $S$. Suppose the contrary. Then there exist not-all-zero scalars $a_1,\ldots,a_m\in F$ such that $$a_1 u_1+\cdots+a_m u_m=\mathbf{0},$$ but the expression $$a_1 u_1+\cdots+a_m u_m+0 u_{m+1}+\cdots+0 u_n=\mathbf{0}$$ will imply linear dependence of $\{u_1,\ldots,u_n\}$ in $V$, violating the fact that it is a basis for $V$.
Does my reasoning work? Thank you.