How to find $\dim W_1$, $\dim W_2$, $\dim W_1+W_2$, $\dim W_1\cap W_2$ for the following spans? Let $W_1=\{(1,1,2,1), (3,1,0,0)\}$ and $W_2=\{(-1,-2,0,1), (-4,-2,-2,-1)\}$
Apparently $\dim W_1=\dim W_2=2$.
For $\dim W_1\cap W_2$, since $(-4,-2,-2,-1)$ can be expressed as $-(1,1,2,1)-(3,1,0,0)$, $\dim W_1\cap W_2=1$. But what if the spans are complicated, how do you find $\dim W_1\cap W_2$. Do you try matrix on each vector and see which ones has a nontrivial solution?
For $\dim W_1+W_2$, how do you know it without calculating $\dim W_1\cap W_2$?
 A: Put the given vectors into the columns of matrices
\begin{align*}
w_1 &=
\left[\begin{array}{rr}
1 & 3 \\
1 & 1 \\
2 & 0 \\
1 & 0
\end{array}\right] &
w_2 &=
\left[\begin{array}{rr}
-1 & -4 \\
-2 & -2 \\
0 & -2 \\
1 & -1
\end{array}\right]
\end{align*}
The dimensions of $W_1$ and $W_2$ are the ranks of the matrices $w_1$ and $w_2$ respectively. Row reducing gives $\DeclareMathOperator{rref}{rref}$
\begin{align*}
\rref(w_1) &=
\left[\begin{array}{rr}
1 & 0 \\
0 & 1 \\
0 & 0 \\
0 & 0
\end{array}\right]
&
\rref(w_2) &=
\left[\begin{array}{rr}
1 & 0 \\
0 & 1 \\
0 & 0 \\
0 & 0
\end{array}\right]
\end{align*}
These reduced row-echelon forms show that both $w_1$ and $w_2$ have rank two. This proves that $\dim W_1=\dim W_2=2$, as you have observed.
Now, the dimension of $W_1+W_2$ is the rank of the matrix
$$
[w_1,w_2]
=
\left[\begin{array}{rrrr}
1 & 3 & -1 & -4 \\
1 & 1 & -2 & -2 \\
2 & 0 & 0 & -2 \\
1 & 0 & 1 & -1
\end{array}\right]
$$
Row reducing gives
$$
\rref[w_1,w_2]=
\left[\begin{array}{rrrr}
1 & 0 & 0 & -1 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
This proves that $\dim (W_1+W_2)=3$. The dimension formula then implies that $\dim(W_1\cap W_2)=1$.
This method also provides bases for the spaces in question. The pivot columns of $\rref[w_1,w_2]$ correspond to the columns in $[w_1,w_2]$ that form a basis for $W_1+W_2$. The "free" columns of $\rref[w_1,w_2]$ correspond to the columns of $[w_1,w_2]$ that span $W_1\cap W_2$. Here, we have $\DeclareMathOperator{Span}{Span}$
\begin{align*}
W_1+W_2 &=
\Span
\left\{
\left[\begin{array}{r}
1 \\
1 \\
2 \\
1
\end{array}\right],
\left[\begin{array}{r}
3 \\
1 \\
0 \\
0
\end{array}\right],
\left[\begin{array}{r}
-1 \\
-2 \\
0 \\
1
\end{array}\right]
\right\}
&
W_1\cap W_1
&= \Span\left\{
\left[\begin{array}{r}
-4 \\
-2 \\
-2 \\
-1
\end{array}\right]
\right\}
\end{align*}
