Null space of a matrix

  • Thread starter kkingkong
  • Start date
  • #1
5
0

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??
 

Answers and Replies

  • #2
35,287
7,140

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.
 
  • #3
107
13
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!
 
  • #4
35,287
7,140
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!
 

Related Threads on Null space of a matrix

  • Last Post
Replies
4
Views
971
  • Last Post
Replies
8
Views
3K
  • Last Post
Replies
8
Views
11K
Replies
5
Views
5K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
9
Views
14K
Replies
15
Views
523
Replies
5
Views
17K
  • Last Post
Replies
10
Views
2K
Replies
9
Views
2K
Top