I am currently reading a book where the author says that the tangent and cotangent bundles $TM$ and $T^*M$ of a manifold $M$ are complexified.
I am not familiar with Complex Manifolds so looked it up on Wikipedia.
Now I just would like to check whether my impression is correct that if $(x^1, \dots, x^n)$ is a local coordinate system and so $\frac{\partial}{\partial x^i} , i = 1, \dots n$ forms a basis for the corresponing local region in the tangent bundle then a vector $v$ in the complexified tangent space $T^C_pM$ at $p \in M$ can be written as \begin{align} v \,\otimes_\mathbb{R} \mathbb{C} &= (\sum_j v^j \frac{\partial}{\partial x^j} \! |_p) \otimes _\mathbb{R} \mathbb{C} = (\sum_j v^j \frac{\partial}{\partial x^j} \! |_p) \otimes _\mathbb{R} (a + ib) \\ &= \sum_j (av^j \frac{\partial}{\partial x^j} \! |_p \otimes 1 + bv^j \frac{\partial}{\partial x^j} \! |_p \otimes i) \\ &= (\, \sum_j av^j \frac{\partial}{\partial x^j} \! |_p ) \otimes 1 + (\, \sum_j bv^j \frac{\partial}{\partial x^j} \! |_p) \otimes i \\ &= av \otimes 1 + bv \otimes i \end{align}
Is this description correct ? I am very new to tensor products so I hope I could use this as a way to check whether I actually understandt how extension of scalars work, thanks for any feedback!