$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes \underbrace{T^*_p M \otimes \dots \otimes T^*_p M}_s \big).$$ If $ (U_a, \varphi_a = (x_a^1 \dots x_a^n)) $ is a local chart of $M$ in $p$, basis of $T_p M$ and $T_p^* M$ are respectively \begin{gather} \left\lbrace \dfrac{\p}{\p x^1_a}, \dots , \dfrac{\p}{\p x^n_a} \right\rbrace,\\ \left\lbrace dx^1_a = \left( \dfrac{\p}{\p x^1_a} \right)^*, \dots , dx^n_a \right\rbrace. \end{gather} and therefore a basis of $T_{r,s}$ is $$ \left\lbrace \dfrac{\p}{\p x^{i_1}_a} \otimes \dots \otimes \dfrac{\p}{\p x^{i_s}_a} \otimes dx^{j_1}_a \otimes \dots \otimes dx^{j_r}_a \right\rbrace_{i_1 \dots i_s \in \lbrace 1 \dots n \rbrace \atop j_1 \dots j_r \in \lbrace 1 \dots n \rbrace} .$$ I want to show that $T_{r,s}$ is a vector bundle on $M$ by proving that $$ \chi_a \left(\sum k_{i_1 \dots i_s, j_1 \dots j_r} \dfrac{\p}{\p x^{i_1}_a} \otimes \dots \otimes \dfrac{\p}{\p x^{i_s}_a} \otimes dx^{j_1}_a \otimes \dots \otimes dx^{j_r}_a \right) := \left( p , k \right),$$ with $k = \lbrace k_{i_1 \dots i_s, j_1 \dots j_r} \rbrace \in \mathbb R^{n^{r+s}},$ is such that $$\chi_a \circ \chi_b^{-1} (p,k) = (p,g_{ab} k),$$ where $g_{ab} : U_a \cap U_b \rightarrow GL(n^{r+s})$.
Now $$ \chi_a \circ \chi_b^{-1}(p, k) = \chi_a \left(\sum k_{i_1 \dots i_s, j_1 \dots j_r} \dfrac{\p}{\p x^{i_1}_b} \otimes \dots \otimes \dfrac{\p}{\p x^{i_s}_b} \otimes dx^{j_1}_b \otimes \dots \otimes dx^{j_r}_b \right) = \chi_a \left(\sum k_{i_1 \dots i_s, j_1 \dots j_r} \underbrace{\dfrac{\p x^{p_1}_a}{\p x^{i_1}_b} \dots \dfrac{\p x^{p_s}_a}{\p x^{i_s}_b} \dfrac{\p x^{j_1}_b}{\p x^{q_1}_a} \dots \dfrac{\p x^{j_r}_b}{\p x^{q_r}_a}} \dfrac{\p}{\p x^{p_1}_a} \otimes \dots \otimes \dfrac{\p}{\p x^{p_s}_a} \otimes dx^{q_1}_a \otimes \dots \otimes dx^{q_r}_a \right).$$ In which way should I arrange $\; \; \; \; \;$ this $\; \; \; \; \;$ to obtain a $g_{ab} \in GL$?