# Null space of a matrix

## Homework Statement

What is the null space of this matrices.
|1 1 0 3 1|
|0 1 -1 0 1|
|1 1 3 0 1|

## The Attempt at a Solution

I reduced it to rref using agumented matrice(each one equals to zero )
|1 0 0 4 0 0|
|0 1 0 -1 1 0|
|0 0 1 -1 0 0|

and i get get x1=-x4 , x2= x4 + -x5, x3 = x4
but then what is the null space??

Mark44
Mentor

## Homework Statement

What is the null space of this matrices.
|1 1 0 3 1|
|0 1 -1 0 1|
|1 1 3 0 1|

## The Attempt at a Solution

I reduced it to rref using agumented matrice(each one equals to zero )
|1 0 0 4 0 0|
|0 1 0 -1 1 0|
|0 0 1 -1 0 0|

and i get get x1=-x4 , x2= x4 + -x5, x3 = x4
but then what is the null space??
Your work so far is OK, but you just need to take it a little further.

Your final matrix, which BTW has an extra column (6th) that isn't needed, represents this system:
Code:
x[SUB]1[/SUB] = -x[SUB]4[/SUB]
x[SUB]2[/SUB] =  x[SUB]4[/SUB] - x[SUB]5[/SUB]
x[SUB]3[/SUB] =  x[SUB]4[/SUB]
x[SUB]4[/SUB] =  x[SUB]4[/SUB]
x[SUB]5[/SUB] =       x[SUB]5[/SUB]
If you sort of squint your eyes at what I wrote, you might be able to see that every vector x in the nullspace can be written as a linear combination of two vectors that are linearly independent.

Remember that the nullspace is the set of all vectors that would send your original system to the zero vector.

If you could find a set of vectors that are independent and send your system to the zero vector, then it seems as though the span of those vectors would be your entire nullspace.

I think you can get it from there!

Mark44
Mentor
Remember that the nullspace is the set of all vectors that would send your original system to the zero vector.
Slight correction: the nullspace is the set of all vectors that your matrix would map to the zero vector.
If you could find a set of vectors that are independent and send your system to the zero vector, then it seems as though the span of those vectors would be your entire nullspace.

I think you can get it from there!