In chapter 3 of this book, it is stated that tangent space $T_x\mathcal{M}$ of a manifold $\mathcal{M}$ can be found using the relation $$ T_x\mathcal{M} = \text{ker}(DF(x)) $$, if $\mathcal{M}$ is defined using a a level set of constant rank using the function $F: \mathcal{E} \to \Bbb{R}^n$
While I intuitively understand this relation, I am confused with the math in the examples.
E.g. Tangent to a sphere $S^{n-1}$ is given as follows: $$ T_{x} S^{n-1}=\operatorname{ker}(\operatorname{DF}(x))=\left\{z \in \mathbb{R}^{n}: x^{T} z+z^{T} x=0\right\}=\left\{z \in \mathbb{R}^{n}: x^{T} z=0\right\} $$
I specifically do not understand how they obtained this term $x^{T} z+z^{T} x=0$; My understanding of $DF(x)$ is as follows: $\left(\dfrac{\partial F}{\partial x_1}, \dfrac{\partial F}{\partial x_2},\dots \right)$ dot product with the coefficients of the tangent vector, say, ${z} = (z_1,z_2,\dots)$. So, we are looking for the $z$ that makes this dot product zero. Hence, I arrive directly at $x^Tz =0$ instead of the intermediate step. I do not understand how this $x^{T} z+z^{T} x=0$ equation is reached.
Q1) Is my approach of looking for the kernel correct? Q2) Which approach leads to this intermediate equation?
Similarly, while finding the tangent space of orthogonal Stiefel Manifold,$$ \operatorname{St}(p, n)=\left\{X \in \mathbb{R}^{n \times p}: X^{T} X=I_{p}\right\} $$, it is stated that kernel is $ \left\{Z \in \mathbb{R}^{n \times p}: X_{0}^{T} Z+Z^{T} X_{0}=0\right\} $. For $\operatorname{St}$ manifold case, evaluating the Jacobian like I did for the $S^{n-1}$ is cumbersome.
Q3) I am not even sure how to evaluate $DF(x)$ and looking for the corresponding $Z$ that makes $DF(X)Z$ zero.