I'm reading Hoffman and Kunze's Linear Algebra and on page 102 of the 2nd edition, they state and prove the following corollary:
Corollary. If $W$ is a $k$-dimensional subspace of an $n$-dimensional space $V$, then $W$ is the intersection of $(n-k)$ hyperspaces in $V$.
Proof. This is a corollary of the proof of Theorem 16 rather than its statement. In the notation of the proof, $W$ is exactly the set of vectors such that $f_i(\alpha) = 0$, $i = k+1,\dots,n$. In case $k = n-1$, $W$ is the null space of $f_n$.
It seems they want to prove $W=\bigcap_{i=k+1}^n \ker f_i$. I understood why $W\subset \ker \bigcap_{i=k+1}^n f_i$, I'm having problems with the converse.