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

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

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

so the vector that spans the Null Space is

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

solving the system Ax=b

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

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

so finally we get all solutions are of the form

$\displaystyle \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ \end{bmatrix}= s \begin{bmatrix} 5 \\ -9 \\ 3 \\ 2 \\ \end{bmatrix}+ \begin{bmatrix} -7 \\ 19 \\ -5 \\ 0 \\ \end{bmatrix}$