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Null space of a matrix

  1. Mar 17, 2014 #1
    1. The problem statement, all variables and given/known data

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


    3. 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??
     
  2. jcsd
  3. Mar 17, 2014 #2

    Mark44

    Staff: Mentor

    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 (Text):
    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.
     
  4. Mar 17, 2014 #3
    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!
     
  5. Mar 17, 2014 #4

    Mark44

    Staff: Mentor

    Slight correction: the nullspace is the set of all vectors that your matrix would map to the zero vector.
     
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