I'm looking at Sergei Winitzki's Linear Algebra via Exterior Products, and he has a question on tensor products.
Firstly we construct the real vector space $\Bbb R^3 \otimes_\Bbb R \Bbb C$ which is 'basically' vectors of the form $v_1 \otimes 1+v_2 \otimes i$ with $v_1,v_2 \in \Bbb R^3$. As a real vector space it is $6$ dimensional.
Then he defines multiplication by a complex number on $\Bbb R^3 \otimes_\Bbb R \Bbb C$ for $\lambda \in \Bbb C$ by
$\lambda(v \otimes z)=v \otimes( \lambda z)$. Then asks what is the dimension of this as a $\Bbb C$ vector space.
I have a feeling that this is now isomorphic to $\Bbb R^3$. The two previously linearly independent vectors $v \otimes 1$ and $v \otimes i$ are now multiples of each other.
I feel like in some sense you're now treating $\Bbb C$ as a vector space over itself in the tensor product now so the dimension is $3\times1$.
I'm not sure if this is the right way to think of these things though.