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Nullity and column spaces

  1. Nov 20, 2008 #1
    Hey fellas could use a hand on a linear question that has been bugging me.

    Give an example of a 4 x 4 matrix such that col A = nul A
    Appreciate any help!
    Thanks,
    ib
     
  2. jcsd
  3. Nov 21, 2008 #2
    To start with, remember that dim(col A) + dim(null A) = 4, so we better have dim(col A) = dim(null A) = 2.
     
  4. Nov 23, 2008 #3
    If col A = null A, then given any vector x in R4, Ax ∈ col A = null A, so A2x = 0. Since x is arbitrary, A2 = 0. Conversely, if A2 = 0, you can show that col A ⊆ null A, and you need dim(col A) = 2. This might help a bit in finding such a matrix.

    (Matrices A such that Ak = 0 for some integer k are called nilpotent.)
     
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