New vector space by combining vectors from the two vector spaces. There are two distinct vector spaces.
What are the ways of making a new vector space by combining vectors from the two vector spaces in different ways? 
 A: I am not really sure what you need but there are a lot of ways to mix vector spaces. The most common are:
-Intersecion:
$U,V$ two K-vector spaces (K field)
$U\cap V= \{v \quad \text{such as} \quad v \in U \quad and \quad v \in V \}$ is a k-space.
-sum:
$U \cup V$ is not an vector space so we define de direct sum $U+V$ as the smallest vector space than contains the sum. In other words, if $U$ Is generated by $S$ And $V$ is generated by the set $T$ then  $U+V$ Is generated by $S\cup T$
-Direct sum:
When $U\cap V=\{0 \}$ then the sum is called Direct sum and its notation is $U \bigoplus V$
-Cartesian product:
The usual Cartesian product of vector spaces is a vector space. In the finite case it is isomorphic to de direct sum.
-Quotient space:
If $V$ is vector space and $U$ is a subspace why can define the vector space $V/U$ where the equivalence relation is : $u \sim v \Leftrightarrow u-v \in U$
Where $[u ]+ [v]= [u+v]$ and $\lambda \in K$ $\lambda[u]=[\lambda u]$
-Dual space:
This case only mix one vector space but maybe is useful for you so if $V$ is a vector space we can define $V^*$ as the space of linear applications from $V$ to $\mathbb{R}$
