I am having trouble with the following example of tangent space. I guess the book is treating tangent vectors as tangent to curves for this part, so I am trying to make sense of it accordingly
Tangent space to an affine subspace
Let $S$ be an affine subspace of $\mathbb{R}^n$ (which is the translation of a vector subspace $W$). The tangent space at a point $p$ of $S$ is $\{p\}\times W$.
In fact, every $(p,\nu) \in\{p\}\times W $ is tangent at $0$ to the curve $p+t\nu$ of $S$, so $\{p\}\times W$ is a subspace of $T_pS$, and therefore $\{p\}\times W$ coincides with $T_p S$
Can someone tell me if the following arguments are correct or explain to me how to do it?
- why is every $(p,\nu) \in\{p\}\times W$ is tangent at $0$ to the curve $p+t\nu$?
Let $\alpha(t)=p+t\nu$, so $\alpha'(t)=\nu$ and $\alpha'(0)=\nu$ . Furthermore $\alpha(0)=p$ So every element of $\{p\}\times W$ is actually a tangent vector $\alpha'(t)$ to the curve $\alpha(t)$ of $S $at $0$ : $(p,\nu)=(\alpha(0), \alpha'(0))$
- how does it follow from here that $\{p\}\times W$ is a subspace of $T_pS$?.
If the previous argument is correct, if every element of $\{p\}\times W$ is a tangent vector to a curve of $S$ of the form $\alpha(t)=p+t\nu$ and if $T_pS$ is the set to all tangent vectors, that is to, any curve in $S$ (not only the ones of the form $\alpha(t)=p+t\nu)$, then $\{p\}\times W \subset T_pS$
- If $\{p\}\times W$ is a subspace of $T_pS$, why should they be equal?
Since $\{p\}\times W \subset T_pS$, and they are both vector spaces, so $\{p\}\times W$ is a subspace of $T_pS$ and since they have the same dimension it follows that they are equal