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A question about (0,0,0) vector

  1. Feb 11, 2008 #1
    when i find that the only vector of ker is (0,0,0)

    what is the dim(ker)

    is it 1
    or is it 0??

    does (0,0,0)
    counts as a vector

    or in case if we find two vectors and one of them is (0,0,0)

    what is the dimention of ker now ??
     
    Last edited: Feb 11, 2008
  2. jcsd
  3. Feb 11, 2008 #2

    EnumaElish

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    Let A be a matrix. By the rank-nullity theorem, rank A + nullity A = rank A + dim(ker A) = number of columns of A, say n(A). So dim(ker A) = n(A) - rank A.

    What is the null space of a full-rank matrix? What is its rank?
     
  4. Feb 11, 2008 #3

    HallsofIvy

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    Staff Emeritus
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    Yes, (0,0,0) is a vector. In fact, since adding (0,0,0) to itself gives, again, (0,0,0) and multiplying (0,0,0) by any number gives, again, (0,0,0), (0,0,0) is the only vector in which the set containing only it is a subspace! Since any vector in a basis must be non-zero, a "basis" for {(0,0,0)} must be empty: its dimension is 0.
     
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