Proving Decomposability of Forms in Spivak's Book (Vol. 1, Chap. 7)

[AiRAVATA]

Sorry to keep bothering, but I am preparing an exam based on Spivak's book on forms (chapter 7 of tome 1).

I need to prove that if $\dim V \le 3$, then every $\omega \in \Lambda^2(V)$ is decomposable, where an element $\omega \in \Lambda^k(V)$ is decomposable if $\omega =\phi_1\wedge\dots\wedge\phi_k$ for some $\phi_i \in V^*=\Lambda^1(V)$.

I think I must use the inner product, but I am not sure. If $\omega \in \Lambda^2(V)$, then

$$\omega=a_{12} \phi_1\wedge \phi_2+a_{13}\phi_1\wedge\phi_2+a_{23}\phi_2\wedge \phi_3$$

I know that if $\{v_1,v_2,v_3\}$ are a basis of $V$, then
$$\begin{array}{l} i_{v_1}\phi_1\wedge\phi_2\wedge\phi_3=\phi_2\wedge\phi_3 \\<br /> i_{v_2}\phi_1\wedge\phi_2\wedge\phi_3=-\phi_1\wedge\phi_3 \\<br /> i_{v_3}\phi_1\wedge\phi_2\wedge\phi_3=\phi_1\wedge\phi_2<br /> \end{array}$$

so

$$\omega=(a_{12}i_{v_3}-a_{13}i_{v_2}+a_{23}i_{v_1}) \phi_1\wedge\phi_2\wedge\phi_3$$

and given the linearity

$$\omega=i_v\phi_1\wedge\phi_2\wedge\phi_3$$

where $v=a_{21}v_1-a_{13}v_2+a_{12}v_3$.

Does that prove the result?

Other idea I had is to express $\phi_i$ in terms of the base of $\Lambda^1(V)$, but I seem to going nowhere.

 

[AiRAVATA]

I think that I've got it.

Let $\phi_1,\phi_2\in \Lambda^1(V)$, where
$$\begin{array}{l}<br /> \phi_1=a_1\varphi_1+a_2\varphi_2+a_3\varphi_3 \\<br /> \phi_2=b_1\varphi_1+b_2\varphi_2+b_3\varphi_3<br /> \end{array}$$

Then

[itex]\phi_1\wedge \phi_2=(a_1b_2-a_2b_1)\varphi_1\wedge\varphi_2+ (a_1b_3-a_3b_1)\varphi_1\wedge\varphi_3+ (a_2b_3-a_3b_2)\varphi_2\wedge\varphi_3[/tex]<br /> <br /> So, given $\omega \in \Lambda^2(V)$, there are (many?) $\phi_1,\phi_2\in\Lambda^1(V)$ such that $\omega=\phi_1\wedge\phi_2$.<br /> <br /> What do you guys think?[/itex]