Suppose $a,b,c$ are in $\Lambda^1 V$. Then I must show that they are linearly dependent iff $a \wedge b \wedge c=0$.
I am new to tensors and wedges and I don't know how to proceed. I could only notice $\Lambda^1 V$ is the dual space $V^*$ so I can treat a,b,c as maps.
Let $a,b,c$ be linearly dependent. Then write $c=ka+rb$. Then $a\wedge b \wedge c(x_1,x_2,x_3)=3! Alt(a \otimes b \otimes c)(x_1,x_2,x_3)=3! \sum_{\sigma \in S_3}a(x_1)b(x_2)(ka+rb)(x_3)= 3! \sum_{\sigma \in S_3}ka(x_1)b(x_2)(a(x_3))+3! \sum_{\sigma \in S_3}ra(x_1)b(x_2)b(x_3) $ I don't seem to be making any progress after this.
Do you know the basic properties of the wedge product? In particular that it is multilinear? So if $c= ka + rb$ then $a\wedge b \wedge c = a\wedge b \wedge(ka+rb) = ka\wedge b\wedge a + r a\wedge b\wedge b=0 $ since $a\wedge b = - b\wedge a$, so whenever $ a$ or $b$ 'occurs' twice in a product that product will vanish.