I have just started learn differential form from the bachman book (page 29)and I found some difficulties in the following problem in 2nd part.
Let $ω(<dx,dy>) = −dx + 4dy$.
1. Compute $ω(<1, 0>)$, $ω(<0, 1>)$ and $ω(<2, 3>)$.
2. What line does ω project vectors onto?
1 st is easy.I need to put values only.
2. here I face the problem.Actually I could not understand what should I need to do and the problem is not clear to me.
seek your help
A $1$-form over $\mathbb R ^n$ (here $n=2$) is an element of the conjugate space $(\mathbb R^n)'$, which is simply $\mathbb R ^n$. Specifcally, if $\omega \in (\mathbb {R} ^n )'$, there exists a unique $w\in \mathbb R ^n$ such that u$$\omega(v)=v\cdot w$$ for each $v\in \mathbb R ^n$ ($\cdot$ is the dot product). You have to find $w$ (actually I think that the correct text of the exercise should be “what vector does omega project vectors onto?”).