Finding the Null Space of a Matrix to Solving for the Solution Set

[kkingkong]

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]

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:

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.

 

[Rellek]

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]

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!