"Suppose that $a,b,c,d$ are linearly independent algebraic $1$-forms on $\mathbb{R}^n$".
What does it mean for algebraic $1$-forms to be linearly independent? I have looked through my notes and cannot find a definition.
An algebraic k-form on a vector space V is function on $V^k$ such that it is linear in each argument and changes sign under the interchange of two arguments. (I think these are tensors of type $(k,0)$).
Hint: Take a linear combination $L=k_1a+k_2b+k_3c+k_4d$ and evaluate at a vector $X$ for which $a(X)=1$ , $b(X)=0$ , $c(X)=0$ and $d(X)=0$.