Concept Clarification: Sum of two subspaces Suppose that $U$ and $W$ are subspaces of $V$. Then the sum $U+W$ is the subspace:
$U+W = \{u+w: u \in U, w \in W\}$
I'm having trouble understand what this means. So if I have the following subspaces $U$ and $W$ of $\mathbb{R}^3$ say:
$U = \{\begin{pmatrix}
x\\ 
y\\
z
\end{pmatrix}: x+y+z=0\}$
$W = \{\begin{pmatrix}
-5x\\ 
3x\\
4x
\end{pmatrix}:$ $x$ is an arbitrary number$\}$
So, if I list some elements of each subspace,
$U = \{\begin{pmatrix}
1\\
1\\
-2
\end{pmatrix}$, $\begin{pmatrix}
2\\
3\\
-5
\end{pmatrix},...\}
$
$W = \{\begin{pmatrix}
-5\\
3\\
4
\end{pmatrix}$, $\begin{pmatrix}
-10\\
6\\
8
\end{pmatrix},...\}
$
Does this mean $U+W = \{\begin{pmatrix}
1\\
1\\
-2
\end{pmatrix} + \begin{pmatrix}
-5\\
3\\
4
\end{pmatrix}$, $\begin{pmatrix}
1\\
1\\
-2
\end{pmatrix} + \begin{pmatrix}
-10\\
6\\
8
\end{pmatrix}$, $\begin{pmatrix}
2\\
3\\
-5
\end{pmatrix} + \begin{pmatrix}
-5\\
3\\
4
\end{pmatrix}$, $\begin{pmatrix}
2\\
3\\
-5
\end{pmatrix} + \begin{pmatrix}
-10\\
6\\
8
\end{pmatrix}, ....
\}
$

What you're writing is exactly correct, although I can see why it would be a bit confusing from this perspective.  It can be tedious to list out infinitely many elements...
A better way to view the space $U+W$ is to write it in formal set notation to get a sense of what an arbitrary vector looks like.
$$U+W=\left\{\begin{pmatrix}a-5d\\b+3d\\c+4d\end{pmatrix}\middle|a,b,c,d\in\mathbb{R},\;a+b+c=0\right\}$$
So now we can see the form that all vectors have in $U+W$.  Now, can you show that any vector in $\mathbb{R}^3$ has this form, so that in fact $U+W=\mathbb{R}^3$?
