[Math] Difference between sum of vector spaces and union of subspaces

Question

I'm having trouble understanding the difference between summing two subspaces and making ther union. My book says that the sum of two subspace is also a subspace, but I've found this example that shows that the union of a subspace is not always a subspace. So, what's the difference?

Answer 1

$V= \mathbb{R}^2, W_1=\langle(0,1)\rangle, W_2=\langle(1,0)\rangle$

Then the union of $W_1$ and $W_2$ is the union of $x$ axis and $y$ axis.

But the sum of them, is all possible combinations of two elements in $W_1,W_2$, thus the sum is $V$, because we can write every element in $V$ as $(x,y)=x(1,0)+y(0,1)$.