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Nullspace and Column Space Question

  1. Nov 21, 2004 #1
    Nullspace and Orthogonal Complement

    Quick question: is the nullspace the orthogonal complement of the column space or the the row space?

    Thanks, sorry I don't have my textbook nearby.
     
    Last edited: Nov 21, 2004
  2. jcsd
  3. Nov 22, 2004 #2
    Hey that was just in my linear algebra exam the other day. The theorem goes:
    nullspace[W]=rowspace[V]

    where W is the subspace and V is the orthogonal complement
     
  4. Nov 22, 2004 #3

    Galileo

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    I never memorize these things. Especially when they are quite easy to reproduce.
    A vector x is in the nullspace of A iff Ax=0.
    The components of Ax are the dot products of the row vectors of A with x. All the components must be zero, so x is in the nullspace of A iff it is orthogonal to all the rows of A.
    Therefore the the nullspace of A the is orthogonal complement of the rowspace of A.
     
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