Let $X$ be a smooth manifold with vector bundles $V$ and $W$. I'm trying to understand how we can construct in general a vector bundle $V \otimes W$.
In particular what manifold structure do we give $V$ and $W$? As a set I understand that: $$V \otimes W = \{ (x,a) : a \in V_x \otimes W_x \}$$
Thanks to Travis' comments my understanding is now as follows:
$V$ and $W$ each have local trivialisation $\Phi_V: \pi_V^{-1}(U) \to U \times \mathbb{R}^k$ and similarly for $W$. Using this we can construct local coordinates on the set $V \otimes W$ above by choosing local coordinates on $U \times \mathbb{R}^k \times \mathbb{R}^l$, however I am struggling to figure out the details of this.
I'm now also wondering how we use the local trivialisation of $V$ and $W$ to construct a local trivialisation for $V \otimes W$?
I would also appreciate it if anyone could recommend a reference for such things - the textbook I am using (Lee) doesn't mention algebraic operations on vector bundles.
Thank you
The confusion here seems to be one of linear algebra, and not of differential geometry. The elements of a tensor product $\Bbb V \otimes \Bbb W$ of two vector spaces can be written as sums $\sum_a v_a \otimes w_a$, and this is enough to describe the manifold structure on the tensor product vector bundle $V \times W \to X$.
Choosing bases $(E_1, \ldots, E_k)$ of $\Bbb V$ and $(F_1, \ldots, F_l)$ of $\Bbb W$ determines a basis $(E_a \otimes F_b)_{1 \leq a \leq k, 1 \leq b \leq l}$ of $\Bbb V \otimes \Bbb W$, which has $\dim(\Bbb V \otimes \Bbb W) = \dim \Bbb V \cdot \dim \Bbb W = kl$ elements. With this basis in hand, we can choose coordinates $(x_i, y_{ab})$ for the vector bundle $V \otimes W$ (which has rank $kl$) with model fiber $\Bbb V \otimes \Bbb W$ by treating the latter as any other vector space for which we've picked a basis.
By contrast, the above choice of bases determine a basis $(E_1, \ldots, E_k, F_1, \ldots, F_k)$ of the direct sum $\Bbb V \oplus \Bbb W$, which has dimension $\dim(\Bbb V \oplus \Bbb W) = \dim \Bbb V + \dim \Bbb W = k + l$.