Scalar extension in tensor product. Let $V$ be a vector space over $\Bbb Q$
a) Show that $\Bbb C$ takes the structure of a $\Bbb Q$ vector space.
b) For elements $\sum'\lambda_i \otimes v_i$ in $\Bbb C \otimes_{\Bbb Q} V$ and $\lambda \in \Bbb C$ define $$\lambda\cdot(\sum'\lambda_i \otimes v_i)=\sum'(\lambda \lambda_i)\otimes v_i$$
Show that $\Bbb C \otimes_{\Bbb Q} V$ has the structure of a $\Bbb C$ vector space
c) Show that for every basis $(v_i)_I$ of V as a $\Bbb Q$ vector space, the family $(1\otimes v_i$ is a basis of $\Bbb C \otimes_{\Bbb Q} V $ as a $\Bbb C$ vector space.
I hate to be someone who just asks for answers, i know this would piss a lot of people off, but after 2 days researching on the internet i still have no idea how to do any of this...Or can someone give me a link where they explain this well?
PS: especially the 3rd question, i dont think i even understand it.
 A: This is essentially an exercise in demonstrating the vector space axioms, in a more complicated and unintuitive setting.
For a) you need to verify that $\Bbb C$ satisfies the vector space axioms when the underlying field is taken to be $\Bbb Q$: we need an abelian group of (vector) addition and for rational multiplication to be compatible (distributive, preserves the identity, etc.) with the (vector) addition structure. Don’t overthink this: this is actually straightforward (rational numbers are themselves elements of $\Bbb C$, which is a field...)
For b), it looks a bit scarier but it’s not. You need to show that the tensor product (the formal sum of the $\otimes$ things) satisfies the vector space axioms when you left-multiply by complex numbers. Conceptually this is more difficult but the real reason why it works is equally as straightforward as part a) (do you even need to worry about the $\otimes V$ part?). Don’t overthink this one either!
For c): we are given a random vector space $V$ whose field of scalars is $\Bbb Q$. We are provided with any basis $(v_i)$ of $V$, which means that the $(v_i)$ are linearly independent in $V$ and span the whole space (under finite linear combination). We need to show that $(1\otimes v_i)$ is also linearly independent under left-multiplication with complex numbers (let’s use the tensor product relations for this, every rational number is also a complex number) and we need to show that they span the whole space, again under left-multiplication by complex numbers. Hint: if I want to obtain $z\otimes v$, I can start by trying to obtain $1\otimes v$ (with my tensor product rules!), and then I can do a multiplication.
