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

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.

• You might need to add a little context to this. What are the $f_i$? And have you looked in the proof of Theorem 16, it seems that this is part of the argument used there.
– SamM
May 12 '16 at 7:31

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 $$\dim W= \dim \text{span}(W\cup \{a\})\implies a\in W\implies \cap_{i=k+1}^n \ker f_i\subseteq W$$
• I didn't understand why the $f_i$ span $(W\cup\{a\})^\circ$. Thank you! May 12 '16 at 12:34
• Well if $a\in \cap_{i=k+1}^n \ker f_i$, then any $g\in (\text{span}(W\cup \{a\}))^\circ$ can be expressed as $g=\sum_{i=k+1}^n g(a_i)f_i$. May 12 '16 at 13:06
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 $\alpha\in W$.