In the context of "The Tangent Space" and after defining "Germs" and to prove that the vector space structure on the tangent space does not depend on the choice of charts, here is the question I am trying to tackle:
Let $X$ be a set, $V,W$ vector spaces, $\alpha: X \to V$ a bijection and $\beta: V \to W$ a linear isomorphism. Then show that the vector space structures induced on $X$ by $\alpha$ and $\beta \alpha$ are equal. Note that $\alpha$ becomes a linear isomorphism with this structure.
My questions are:
1- what are $\alpha$ and $\beta$? are they curves? charts? or what?
2- What exactly should I prove? there was a hint at the back of the book(A short course in differential topology by Bjorn Ian Dundas) saying $$\alpha^{-1}(a \alpha(x) + b \alpha(y)) = \alpha^{-1} \beta^{-1} \beta (a \alpha(x) + b \alpha(y)) = (\beta \alpha)^{-1} \beta (a \beta \alpha(x) + b \beta \alpha(y))$$ but I did not understand exactly what are $x,y$? what exactly this hint is doing and what is the remaining that I should prove and how? where are the rest of the axioms of a vector space that should be proved?
Thanks
So first of all, given your set $X$ you want to install a $k$-vector space structure on it, with some field $k$. That means precisely, you need to define two maps of the form $$ X\times X\to X,~(x,x')\mapsto x+x', $$ which will be called the addition, and $$ k\times X\to X,~(\lambda,x)\mapsto \lambda\cdot x, $$ which will be called scalar multiplication. A priori, these operations do not make sense on the $set$ $X$.
Now you have given a bijection $\alpha:X\to V$ which maps from $X$ into some $k$-vector space $V$. This allows you to pull back the vector space operations of $V$ to define such operations on $X$, namely by setting $$ x+x':=\alpha^{-1}\big(\alpha(x)+\alpha(x')\big)~~~~~~~~~~\text{and}~~~~~~~~~~\lambda\cdot x:=\alpha^{-1}\big(\lambda\cdot\alpha(x)\big), $$ for $x,x'\in X$, $\lambda\in k$. Note that here, on the left-hand sides, $+$ and $\cdot$ are the operations on $X$ which you want to define, and on the right-hand sides, $+$ and $\cdot$ are the operations on $V$ which already exist by the assumption of $V$ being a vector space. $\alpha(x)$ and $\alpha(x')$ are now vectors in $V$ which may be operated by $V$'s vector space operations.
Now you have given an isomorphism $\beta:V\to W$ and thereby, in particular, another bijection $\beta\circ\alpha:X\to W$ defined on the set $X$, mapping into some $k$-vector space (now $W$). With this bijection $\beta\circ\alpha$ you can now do the analog construction as above and obtain operations $+$ and $\cdot$ which turn $X$ into a $k$-vector space. So now, you have two notions of $+$ and two notions of $\cdot$ using $\alpha$ and $\beta\circ\alpha$, respectively, which are a priori different operations.
Your problem may now be written as: "Show that the two pairs of operations in fact coincide." Formally, you need to show $$ x+x'\Big|_{\text{using }\alpha}=x+x'\Big|_{\text{using }\beta\circ\alpha},~~~~~\text{i.e.}~~~~~\alpha^{-1}\big(\alpha(x)+\alpha(x')\big)=\big(\beta\!\circ\!\alpha\big)^{-1}\big(\beta\!\circ\!\alpha(x)+\beta\!\circ\!\alpha(x')\big) $$ for any choice of $x,x'$, and accordingly for $\cdot$. This is essentially what the book's hint says.