Find all possible solutions to W1 and W2.
Find the set of all vectors which are solutions to both systems and why it is a vector space.
Please help.
The augmented matrix for the first is
$\displaystyle \begin{bmatrix}
1 && 2 && 1 && 0 \\
2 && 4 && 2 && 0 \\
1 && 2 && 1 && 0 \\
\end{bmatrix}
$
In reduced tow eschelon form you get...
$\displaystyle \begin{bmatrix}
1 && 2 && 1 && 0 \\
0 && 0 && 0 && 0 \\
0 && 0 && 0 && 0 \\
\end{bmatrix}
$
so we have two "free" variables we get the solution
$\displaystyle x=-2s-t$
$\displaystyle y= s$
$\displaystyle z=t $
$\displaystyle v_=\begin{bmatrix}
-2s-t \\
s \\
t
\end{bmatrix}= s\begin{bmatrix}
-2 \\
1 \\
0 \\
\end{bmatrix}+ t\begin{bmatrix}
-1 \\
0 \\
1 \\
\end{bmatrix}$
The last two vectors span the solution space of the first matrix.
Try this proceedure with the second matrix and compare the solution spaces.
I hope this helps
Good luck.