I am trying to see if I understand the idea of a bases as presented in the text by Friedberg, Insel, and Spence. Please correct any and all misconceptions I have in the following and feel free to add insight if necessary:
1.) The definition of the bases of a vector space is a linearly independent set that generates the vector space. The number of vectors in the base are always the same, where when I say "same" I mean that there can be a one to one correspondence between the two bases. So even in an infinite dimensional vector space, the bases will have the "same" number of vectors"
2.) I find the following exercise too trivial for it to be correct: Prove that a vector space is infinite dimensional if and only if it contains an infinite linearly independent set.
If a vector space $V$ is infinite dimensional, then by definition it has a bases of infinite vectors. On the other hand, if a vector space has an infinite linearly independent set S, then $Span(S)\subset V$ but $Span(S)$ has infinite dimension so $V$ must also.
3.) If two bases are given of the "same" size, then the space generated by them will be of equal "size." Specifically, I will be able to find some 1-1 correspondence between the two spaces that are generated.
4.) Heuristically, all we have to do to find the dimension of a finite dimensional vector space is consider the number of different variables needed to express an element of the space.
5.) If $H \subset W$, where $H$ and $W$ are subspaces of $V$ with bases $\alpha$ and $\beta$, then, even in the infinite case, $\alpha \subset \beta$. In particular $(\alpha \cap \beta) \subset \alpha$ and $\alpha \subset (\alpha \cup \beta)$. Does everything here remain true even when we extend to infinite intersections (and unions). Also what are conditions for $\alpha \cap \beta$ to be a basis for $H \cap W$ (same with union)?