I am struggling with part of a proof about the normal space of a submanifold. The text states:
Lemma: Suppose $M = f^{-1}(q) \subset \mathbb{R}^{n+k}$ is an $n$-dimensional submanifold. Then for the normal space of $M$ at $p$ we have $N_p M = span \{grad f_1(p),...,grad f_k(p)\}$.
The proof starts like this:
Proof: To exhibit $N_p M$, regard the matrix product $J_f \cdot v$ as a scalar product with the rows of the Jacobian:
\begin{equation*} J_f \cdot v= \begin{pmatrix} (grad f_1)^T \\ \vdots \\ (grad f_k)^T \end{pmatrix} \cdot v = \begin{pmatrix} \langle grad f_1,v \rangle \\ \vdots \\ \langle grad f_k,v \rangle \end{pmatrix} \end{equation*}
Choosing for $v$ one of the gradient vectors grad $f_i$ results in a nonzero product, that is $span \{grad f_1(p),...,grad f_k(p)\} \perp$ ker $J_f(p)$. In general, $Y \perp X$ implies $Y \subset X^{\perp}$. Therefore, in our case,
$span \{grad f_1(p),...,grad f_k(p)\} \subset (ker \ J_f)^{\perp} = (T_p M)^{\perp} = N_p M$.
But both sides are $k$-dimensional vector spaces, and so the spaces must be equal.
The definition of a submanifold I am using is:
Definition: A non-empty subset $M \subset \mathbb{R}^{n+k}$ is called an $n$-dimensional $C^{\alpha}$-submanifold if $M = f^{-1}(q)$ where $f \in C^{\alpha}(\mathbb{R}^{n+k},\mathbb{R}^{k})$ has $q \in \mathbb{R}^{k}$ as a regular value, and $\alpha \in \mathbb{N}$.
Now I understand that if $v=f_i$ for some $i$, then $J_f \cdot v \neq 0$ since $J_f$ has rank $k$ which means that the rows of $J_f$ are all non-zero. This means that the $i$th row of $J_f \cdot v$ is $\|grad f_i\| \neq 0$. However, I don't see why this implies that $span \{f_1(p),...,f_k(p)\}$ is orthogonal to the kernel. Of course, from the observation above we know that none of the $f_i$ is an element of the kernel though.
What am I missing here? I feel like this is something easy. Appreciate if someone can point me in the right direction.
Thanks a lot!