Null Space

**Bloden** · November 29th 2006, 12:04 PM

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).

**galactus** · November 30th 2006, 08:03 AM

$\begin{bmatrix}1&3&5&0\\0&1&4&-2\end{bmatrix}$

Find your rref:

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

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

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

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

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

Therefore, Null(A) =

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

and

$\begin{bmatrix}-6\\2\\0\\1\end{bmatrix}$

**Bloden** · November 30th 2006, 09:15 AM

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.

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