Linear Space in Vector Spaces question. 
How do I do the first part of the question where they say V1 U V2 is not a linear space, please help my exam is very close.
In the marking scheme it says it's not closed under addition but can someone explain me why , or give me an example why it's not closed under addition.
I did it this way -let s= c (b1) + c2 (b2) + c3(b3) + c4(b4) , then I took v= a1 (b1) + a2 (b2) + a3 (b3) + a4 (b4) , then added to give v+s = (c+a1) b1 + (c2+a2)b2.........and adding it separately , it shows they r closed under addition , please help me in this , btw like this is the first time I have seen such a question In my exam board becoz normally they only ask for the rank , # of dimensions etc , this type I'm really not used to at all. I can't find something that is not closed under addition.
 A: Here's some geometric intuition about what is going on.
It's pretty unlikely that the union of two linear spaces could be a linear space. Think about two lines in 2D:
$$V_1 = \text{span}(\begin{bmatrix}1 \\ 0\end{bmatrix}), \quad V_2 = \text{span}(\begin{bmatrix}1 \\ 1\end{bmatrix})$$
the union of them $V_1 \cup V_2$ makes a cross, which looks as follows:

This orange cross can't be a linear space because if you add a vector in one of the lines in the cross to a vector in the other line of the cross, you get something like the blue points which are not in the cross anymore. The existence of these blue points outside the cross demonstrates that $V_1 \cup V_2$ is not closed under addition.

Now let's take it up a level: two planes in 3D:
$$V_1 = \text{span}(\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}, \begin{bmatrix}1 \\ 1 \\ 0\end{bmatrix}), \quad V_2 = \text{span}(\begin{bmatrix}1 \\ 1 \\ 0\end{bmatrix}, \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix})$$
The union of these two spaces $V_1 \cup V_2$ is the union of two planes that cut through each other, which look as follows:

Again we see that there could be an infinity of (blue) points that are combinations of points on the orange planes, but that themselves are not on either plane.
A: Guide to getting to the solution


*

*Write down what it takes to be a linear space. This of often presented as a list of 10 requirements, or axioms.

*As this is actually a subset of a linear space, we wonder whether $V_1 \cup V_2$ is a "linear subspace." While this might sound like it makes things harder, because we've introduced another definition (subspace), in fact it makes many things simpler, as we know that $V_1 \cup V_2$ has many properties simply because it's in $\mathbb{R}^4$. So write down what you need to check to see if it is a linear subspace.

*It's often easiest to try to find a counterexample (if there is a counterexample). So choose some vectors in your space. Choose, say, 10 or 15 pairs if necessary, and check what you expect from these pairs of vectors from #2 above. Since you are given these subspaces in terms of bases, choosing basis vectors is often a good choice. 

*You're correct: it's not closed under addition. So in particular, there are pairs of vectors that you can choose which fail an additive property.

