14.22: pg 362 Lee's Smooth Manifold Let $X$ be a smooth vector field on $M$.
If $w$ is a smooth differential form then $i_Xw$ is smooth.
$$ i_X w:= (X \lrcorner w)_p =X_p \lrcorner w_p. $$$i_X:\Omega^k(M) \rightarrow \Omega^{k-1}(M)$ is linear over $C^\infty(M)$, corresponding to a smooth bundle homomoprhism $$ i_X: \wedge ^kT^*M \rightarrow \wedge^{k-1}T^*M. $$
I would really appreciate nice explanations for both aprts. My solution for 1. is ugly (below). For 2. how does linearity over $C^\infty(M)$ imply bundle homomorphism?
My thoughts for 2. are using lemma 14.13 and working in a local coordinate chart.
Lemma 14.13, pg 358: If $w \in \wedge^k(V^*) $ and $\eta \in \wedge^l(V^*)$, $$ i_v (w \wedge \eta) = (i_v w ) \wedge \eta + (-1)^k w \wedge (i_v \eta). $$
which I believe is what the solution wants.
May attempt for 1.
We work locally on a local chart $(U, (x^i))$ around $w$. We know $w = \sum w_I dx^I$, $X=X^i \frac{\partial}{\partial x^i} $, where $w_I, X^i$ are smooth on $U$. Thus, for $p \in U$, $v_2, \ldots, v_k \in T^kM$, \begin{align*} X_p \lrcorner w_p (v_2, \ldots, v_k) &= w_p(X_p, v_2, \ldots, v_k) \\ &= \sum_I w_I(p) \sum_i X^i (p)dx^I(\frac{\partial }{\partial x^i}, v_2, \ldots, v_k) \end{align*}
In this form, we can see the expression is clearly representable $\sum_{J} l_Jdx^J$ for $J$ of length $k-1$., such that $l_J$ is smooth on $U$.
EDIT:
Thanks for the comments below. The general theorem show that $i_X$ is then given by $i_X (v) = i_X (\tilde{v})(p)$ for some $\tilde{v} \in \Omega^k(M)$ where $\tilde{v}(p) = v$.