# Null Space

• Nov 29th 2006, 12:04 PM
Bloden
Null Space
Given A = [[1,3,5,0],[0,1,4,-2]] (where A is a 2x4 matrix, incase you dont know my notation):

Find a basis for the nul(A). Then, describe the geometry of the nul(A).
• Nov 30th 2006, 08:03 AM
galactus
$\displaystyle \begin{bmatrix}1&3&5&0\\0&1&4&-2\end{bmatrix}$

$\displaystyle \begin{bmatrix}1&0&-7&6\\0&1&4&-2\end{bmatrix}$

The nullspace is the solution space to the system of homogenoeous equations Ax=0.

$\displaystyle x_{1}-7x_{3}+6x_{4}=0$
$\displaystyle x_{2}+4x_{3}-2x_{4}=0$

$\displaystyle x_{1}=7x_{3}-6x_{4}$
$\displaystyle x_{2}=-4x_{3}+2x_{4}$

$\displaystyle x_{1}=7s-6t$
$\displaystyle x_{2}=-4s+2t$
$\displaystyle x_{3}=s$
$\displaystyle x_{4}=t$

Therefore, Null(A) =

$\displaystyle \begin{bmatrix}7\\-4\\1\\0\end{bmatrix}$

and

$\displaystyle \begin{bmatrix}-6\\2\\0\\1\end{bmatrix}$
• Nov 30th 2006, 09:15 AM
Bloden
I just don't get what the "basis" for the null is; the linear combinations? We can automatically tell that it's lin. dependent because we can't get pivots in every row, so how do you get a lin combo.