About the $\dim(\{\phi\in V^*:\phi\wedge T=0\})$

Question

Hi I need help with this exercise: Let $V$ a vector space such that $\dim (V)=n$ and $n(T)=\{\phi\in V^*:\phi\wedge T=0\}$. I want to show that if $T\in\wedge^{n-1}V^*$ then $\dim(n(T))=n-1$. Thanks.

Answer 1

I guess you are assuming $T\neq0$. You can write everything in coordinates. Let $dx^1,\ldots,dx^n$ be a basis of $V^*$. So, write $$T=\sum T_idx^1\wedge\ldots\widehat{dx^i}\ldots\wedge dx^n,$$and then $\phi\wedge T=0$ becomes a homogeneous linear equation in $\phi_1,\ldots,\phi_n$, where $$\phi=\phi_idx^i.$$