1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Nullspaces of a Matrix

  1. Feb 11, 2012 #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. Feb 12, 2012 #2

    HallsofIvy

    User Avatar
    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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Nullspaces of a Matrix
  1. Matrix ? (Replies: 4)

  2. Matrix question (Replies: 7)

  3. Matrix Problem (Replies: 5)

Loading...