Suppose an inner product on an $n$-dimensional vector space $V$ is fixed and $\{e_1, e_2, \cdots, e_n\}$ is an orthonormal basis for $V$. For any $k$, define an inner product $\langle\cdot, \cdot \rangle$on $A_k$ which is determined uniquely by the property that the basis element $\{\alpha_I\}_I$ form an orthonormal basis. Here $I$ ranges over all $I=(i_1, \cdots, i_k)$ such that $1\leq i_1<i_2<\cdots<i_k\leq n$. Let $\omega=\alpha_1\wedge\cdots\wedge \alpha_n$.
(1) For any $\alpha\in A_k(V)$, there is a unique element $\ast_k\alpha\in A_{n-k}(V)$ such that $\beta\wedge\ast_k\alpha=\langle\beta, \alpha\rangle\omega$, and the map $\ast_k:A_k(V)\rightarrow A_{n-k}(V)$ is linear.
(2) $\ast_{n-k}\circ\ast_k=(-1)^{k(n-k)}$.
(3) Suppose $M$ is a smooth $n$-dimensional manifold with a smooth inner product on its tangent space. Then the linear operator induces a linear map $\ast_k:\Omega^k(M)\rightarrow \Omega^{n-k} (M)$.
Related Notes on (3): Suppose $M$ is a compact smooth manifold, $(E, M, \pi)$ is a smooth vector bundle. We write $E_p$ for the fiber of this vector bundle over $p\in M$. A smooth inner product on $E$ consists of a symmetric bilinear form $g_p:E_p\times E_p\rightarrow\mathbb{R}$ for each $p\in M$ such that $g_p(v,v)>0$ for any non-zero $v\in E_p$, and for any pairs of smooth sections $s_1$ and $s_2$ of $E$ $\langle s_1, s_2\rangle(p):=g_p(s_1(p), g_2(p) )$ defines a smooth function on $M$.
What I know: There always exists a smooth inner product on $E$. And $TM$ and $T^\ast M$ are isomorphic vector bundles over $M$.
(4) Suppose $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a smooth function. Then what is a formula for $\ast_nd\ast_1df=0$?
(5) What is a global frame for $\Omega^2(\mathbb{R}^4)$ such that any element $\omega$ of this frame satisfies $\ast_2\omega=\omega$ or $\ast_2\omega=-\omega$?
I got part (1) and (2). But I need a help on (3), (4) and (5). I am studying for a qual. Any help is greatly appreciated! Thank you.