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Non-zero solution of a homogenous system

  1. Mar 27, 2014 #1
    Hi,
    Could anyone give me a proof for the following theorem?
    Theorem : Ax=0 has a non-trivial solution iff det(A)=0

    Thanks in advance.
    /Sunny
     
  2. jcsd
  3. Mar 27, 2014 #2

    micromass

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    You will find the proof in books such as "Linear Algebra" by Lang. I suggest you check out that book.
     
  4. Mar 27, 2014 #3
    This is easier to understand if you think about the geometric meaning of a determinant.

    A matrix [itex]A[/itex] represents a linear transformation in a given basis. The determinant is a property of the matrix, but it's also a property of the linear transformation it represents, independently of any basis. The determinant is simply the signed volume change of the volume element.

    Now we can see what a zero determinant means: it means the transformed volume element is zero. This transformation has squished it down into a lower-dimensional subspace, which means the mapped basis vectors can't be linearly independent. That means there's a nontrivial linear combination of basis vectors which maps to 0. In other words, there must be some [itex]x[/itex] in this basis such that [itex]Ax = 0[/itex].
     
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