Nullspaces of a Matrix

  1. 1. The problem statement, all variables and given/known data

    Given an mxn matrix M, prove that Null((M^T)(M)) = Null(M)

    Where M^T is the transpose of the matrix M.

    3. The attempt at a solution

    I was able to get the first part (Null(M) is a subset of Null((M^T)(M))), but I'm just having trouble proving the other way around. I pick any vector in Null((M^T)(M)), but unsure of what to do after that.
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,946
    Staff Emeritus
    Science Advisor

    The standard way to prove that to sets, A and B, say, are equal is to prove A is a subset of B then prove that B is a subset of A. And you prove A is a subset of B by starting "if x is in A" and then use the properties of A and B to conclude "x is in B".

    Here, the two sets are Null(M) and Null(M^T(M)). If x is in Null(M) then M(x)= 0. It then follows immediately that M^T(Mx)= M^T(0)= 0. That's the easy way. If x is in Null(M^T(M)) then M^T(M(x))= 0. Obviously, if M(x)= 0 we are done. What can you say about non-zero x such that M^t(x)= 0?
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