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Finding rank(range) and nullspace of a matrix

  1. Jun 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Trying to figure out the rank and nullspace of the matrix of matrix A and B:

    A=
    1 0
    5 4
    1 4

    B=
    1 0 1
    5 4 9
    2 4 6
    2. Relevant equations
    I used the Guass elimination on both


    3. The attempt at a solution

    For A I said r3[itex]\rightarrow[/itex]r3-r1, then r3→r3+4r1 then r2→r2-5r1 that lead to me getting
    A=
    1 0
    0 4 Rank=2 and Null space=0
    0 0

    For B I said r3→r3-r2, then r3→r3+3r1 that lead me to:
    B=
    1 0 1
    5 4 9 Rank=2 and Null space=1
    0 0 0

    Am I on the right track or do I have these completly wrong?
     
  2. jcsd
  3. Jun 13, 2012 #2

    Simon Bridge

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  4. Jun 13, 2012 #3

    vela

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    Are you trying to find the null space or the nullity, which is the dimension of the null space?
     
  5. Jun 13, 2012 #4
    I am looking for the nullspace not the nullity. The "nullspace" that I have in the fist post is the nullity. I have been stuyding this and am using the formula Axp=c and am not really understanding how I got a nullspace of (0,0)

    I found A=
    1x1+0x2=0
    0x1+4x2=0
    0x1+0x2=0

    and a nullspace of:
    0
    0

    still looking at B
     
  6. Jun 13, 2012 #5

    vela

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    The null space is just the set of vectors that satisfy Ax=0. In your first example, the only solution is x=(0,0), so the null space is {(0,0)}, which is a vector space of dimension 0.

    For your second problem, you found the nullity is 1, so the null space should turn out to be a vector space of dimension 1. That is, it should be the multiples of some vector. You want to figure out what that vector is by solving Bx=0.
     
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