Help me with basis and dimension Suppose that $A$ and $B$ are vector subspaces of $V$, and let $C$ and $D$ be bases for $A$ and $B$, respectively.
Then is it true that 


*

*$C \cup D$ is a basis for $A+B$?

*$\operatorname{dim}(A+B) \le |C| + |D|$ (where $|\cdot|$ denotes cardinality)?


I am really bad at dimensions and bases.
Can you help me to understand such questions and how to deal with them?
 A: Consider $\Bbb{R}$ as $\Bbb{R}$-vectorspace. Two basis are given by $C:=\{1\}, D:=\{2\}$. Can you answer your first question from here?
For the second question notice that $C\cup D$ is a generating system for $A+B$.
A: In questions like this, it's always good to consider cases where $A$ and $B$ have non-trivial intersection, as well as cases where they intersect trivially.
Suppose $V=\Bbb R^3$, and take $C=\{(1,0,0),(0,1,0)\}$ and $D=\{(0,1,1), (0,0,1)\}$. Do you see how this furnishes a counterexample for (1)?
For (2), all you need to show is that $C\cup D$ spans $A+B$. Can you write an arbitrary element of $A+B$ in terms of basis vectors of $A$ and $B$?
A: There is a linear map $\;\begin{aligned}[t]f:A\times B&\longrightarrow A+B\\ (u,v)&\longmapsto u-v\end{aligned}$. 
The kernel of $f$ is the subspace
$\;\bigl\{(u,v)\in A\times B\mid u=v\bigr\}$, hence it is isomorphic to $A\cap B\;$ by the linear map:
$$A\cap B\longrightarrow  A\times B,\enspace u\longmapsto (u,u).$$
Now $A\times B$ has dimension equal to $\dim A+\dim B$. Indeed, a basis of $A\times B$ is the union of   $C'=C\times\{0\}$ and $D'=\{0\}\times D$.
By the rank-nullity theorem, we have
$$\dim(A+B)=\dim A+\dim B - \dim A\cap B,$$
and we see $C\cup B$ is a basis for $A+B$ if and only if $A\cap B=\{0\}$, i.e. if and only if the sum $A+B$ is direct.
