Orientation of Vector Space Arising From a Vector

Question

I'm self-studying Gullemin's "Differential Forms." It's very interesting but, for me at least, challenging not to have the benefits of a professor or lectures. I'm having trouble understanding the material in Section 1.9, which concerns orientations of vector spaces. In particular, I have a question about the last part of problem 1.9.vi:

"Exercise 1.9.vi. Let $V$ be an oriented $n$-dimensional vector space and $W$ an $(n-1)$-dimensional subspace. Show that if $v$ and $v'$ are in $V \backslash W$ then $v' = \lambda v + \omega$ where $\omega$ is in $W$ and $\lambda \in \mathbb{R} \backslash \{0\}$. Show that $v$ and $v'$ give rise to the same orientation of $W$ if and only if $\lambda$ is positive."

I've solved the first part (i.e., finding $\lambda$ and $\omega$ such that $v' = \lambda v + \omega$). I don't understand the last sentence. I've re-read Section 1.9 several times, and I'm not clear how vectors $v$ and $v'$ give rise to an orientation of $W$. Can someone please explain or point me to an online resource that will clarify this? Note that I'm not looking for a solution (although feel free to provide hints), just enough information so I understand the problem.

For context, you can find an early draft of this book at https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf. See page 43 of the PDF.

Many thanks for your assistance!

Best, Doug

Answer 1

An orientation is the choice of an $n$ alternated form $\omega$, the orientation induced by $v$ on $W$ is the $n-1$ form $\omega_v(w_1,..,w_{n-1})=\omega(v,w_1,..,w_{n-1})$, suppose that $v'=\lambda v+w, \lambda>0$, $\omega_{v'}(w_1,..,w_{n-1})=\omega(\lambda v+w,w_1,..,w_{n-1})=\lambda\omega(v,w_1,..,w_{n-1})=\lambda \omega_v(w_1,..,w_{n-1})$.

We deduce that $\omega_{v'}=\lambda\omega_v$ which is equivalent to saying that $\omega_v$ and $\omega_{v'}$ define the same orientation if and only if $\lambda>0$.

Answer 2

My guess of what they mean is that since $v\in V\setminus W$, it is a non-zero vector. Now, we can choose an ordered basis $\beta:= \{w_1, \dots, w_{n-1}\}$ of $W$, and then we can extend it to an ordered basis $B_v:= \{v, w_1, \dots, w_n\}$ a basis of $V$. The basis $B_v$ could be either positively oriented or it could be negatively oriented (with respect to the given orientation on $V$). We shall now prescribe an orientation on $W$ by saying (the equivalence class containing) $\beta$ is positively-oriented if and only if $B_v$ is a positively oriented basis.

Similarly, using $v'$, we get another basis $B_{v'}$, which then induces an orientation on $W$.