# Math Help - Consider 2 homogeneous systems

1. ## Consider 2 homogeneous systems

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.

2. The augmented matrix for the first is

$\begin{bmatrix}
1 && 2 && 1 && 0 \\
2 && 4 && 2 && 0 \\
1 && 2 && 1 && 0 \\
\end{bmatrix}
$

In reduced tow eschelon form you get...

$\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

$x=-2s-t$
$y= s$
$z=t$

$v_=\begin{bmatrix}
-2s-t \\
s \\
t
\end{bmatrix}= s\begin{bmatrix}
-2 \\
1 \\
0 \\
\end{bmatrix}+ t\begin{bmatrix}
-1 \\
0 \\
1 \\
\end{bmatrix}$

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.

3. thanks man.
I got the following solution for second one.
So is S[-2，1，0] the solution for both systems?