# Math Help - Nullspace

1. ## Nullspace

Ax=b
a) calculate the nullspace Null(A) and
b) give the general solution of the system Ax=b or state that the system inconsistent.
For b) write in the form x= p+y where p is a particular solution and y ranges across Null (A)

A=1 1 0 2
2 1 1 -2
2 2 2 1

b= 12
0
14

2. Originally Posted by Rui
Ax=b
a) calculate the nullspace Null(A) and
b) give the general solution of the system Ax=b or state that the system inconsistent.
For b) write in the form x= p+y where p is a particular solution and y ranges across Null (A)

A=1 1 0 2
2 1 1 -2
2 2 2 1

b= 12
0
14
To find the Null space solve the matrix equation Ax=0

$\begin{bmatrix}
1 && 1 && 0 && 2 && 0 \\
2 && 1 && 1 && -2 && 0\\
2 && 2 && 2 && 1 && 0 \\
\end{bmatrix} = \begin{bmatrix}
1 && 0 && 0 && -5/2 && 0 \\
0 && 1 && 0 && 9/2 && 0\\
0 && 0 && 1 && -3/2 && 0 \\
\end{bmatrix}$

now is $x_4=t$ then $x_3=3t/2$
$x_2=-9t/2$ $x_1=5t/2$

so the vector that spans the Null Space is

$v= \begin{bmatrix}
5t/2 \\
-9t/2 \\
3t/2 \\
t \\
\end{bmatrix} = t/2 \begin{bmatrix}
5 \\
-9 \\
3 \\
2 \\

\end{bmatrix} = s \begin{bmatrix}
5 \\
-9 \\
3 \\
2 \\

\end{bmatrix}

$

solving the system Ax=b

$\begin{bmatrix}
1 && 1 && 0 && 12 && 0 \\
2 && 1 && 1 && 0 && 0\\
2 && 2 && 2 && 14 && 0 \\
\end{bmatrix} = \begin{bmatrix}
1 && 0 && 0 && -5/2 && -7 \\
0 && 1 && 0 && 9/2 && 19\\
0 && 0 && 1 && -3/2 && -5 \\
\end{bmatrix}$

A particualr solution is
$\begin{bmatrix}
-7 \\
19 \\
-5 \\
0 \\
\end{bmatrix}$

so finally we get all solutions are of the form

$\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
\end{bmatrix}= s \begin{bmatrix}
5 \\
-9 \\
3 \\
2 \\

\end{bmatrix}+ \begin{bmatrix}
-7 \\
19 \\
-5 \\
0 \\
\end{bmatrix}
$