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Invertible Matrix

  1. Apr 12, 2009 #1
    1. The problem statement, all variables and given/known data

    If A is a an mxn matrix and its column vectors are linearly independent.

    Prove that the matrix AtA is nonsingular. Hint: Use the fact that it is sufficient to show that null(AtA) = {0}

    2. Relevant equations



    3. The attempt at a solution

    I'm new to this topic & I don't understand the hint given and how exactly to use it to prove the question...

    I know that in order for a matrix to be nonsingular/invertible it has to be squre (m=n) and when you multiply the a matrix by its transpose, the resulting matrix would be square.
    I'm also thinking about the properties of fundamental spaces of matrices that: row(A) = null(A) and col(A)=null(At) (therefore null(AtA) = row(A).col(A)?)

    Any help would be much appreciated :)

    Cheers.
     
  2. jcsd
  3. Apr 12, 2009 #2

    Dick

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    If the columns of A are linearly independent, doesn't that mean null(A)={0}? Now suppose A^(T)A were singular and think about x^(T)A^(T)*Ax, where x is a column vector in R^n.
     
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