rref:
\begin{bmatrix}
1 & 0 & -1 & -2 & -3 \\
0 & 1 & 2 & 3 & 4 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
Homogeneous System:
v-x-2y-3z = 0 \\
w+2x+3y+4z = 0
- Find a basis for the column space of M.
\begin{bmatrix}
1 \\
0 \\
0 \\
\end{bmatrix}
\begin{bmatrix}
0 \\
1 \\
0 \\
\end{bmatrix}
- Find a basis for the null space of M.
v-x-2y-3z = 0 \\
v = x-+2y+3z\\
\\
w+2x+3y+4z = 0\\
w = -2x-3y-4z\\
\\
x = x\\
y = y\\
z = z
Thus:
\begin{bmatrix}
v \\
w \\
x \\
y \\
z
\end{bmatrix}
=x
\begin{bmatrix}
1 \\
-2 \\
1 \\
0 \\
0
\end{bmatrix}
y
\begin{bmatrix}
2 \\
-3 \\
0 \\
1 \\
0
\end{bmatrix}
z
\begin{bmatrix}
3 \\
-4 \\
0 \\
0 \\
1
\end{bmatrix}
