Help in this corollary in Hoffman and Kunze's linear algebra book

Question

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.

Answer 1

Suppose $a\in \cap_{i=k+1}^n \ker f_i$. Then the $f_i$ also span $(\text{span}(W\cup \{a\}))^\circ$, giving us $\dim W^\circ= \dim (\text{span}(W\cup \{a\}))^\circ$. Hence $$\dim W= \dim \text{span}(W\cup \{a\})\implies a\in W\implies \cap_{i=k+1}^n \ker f_i\subseteq W$$

Answer 2

Suppose $\alpha\in \cap_{i=k+1}^n \ker f_i$. Using the notation of the book and by (3-14) on page 99, we know that $\alpha=\sum_{i=1}^nf_i(\alpha)\alpha_i$. Therefore $\alpha=\sum_{i=1}^kf_i(\alpha)\alpha_i$, so $\alpha\in W$.