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(Linear Algebra) Vector Space

  1. Oct 21, 2010 #1
    1. The problem statement, all variables and given/known data
    The set of all 2x2 singular matrices is not a vector space. why?
    [tex]\begin{bmatrix} 1 & 0\\ 0&0 \end{bmatrix}+\begin{bmatrix} 0 & 1\\ 0& 1 \end{bmatrix}=\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}[/tex]

    2. Relevant equations
    Is it because the determinant in both are zero, but by performing addition you get a nonsingular matrix from a two singular matrices.


    3. The attempt at a solution
    det(0)+det(0)=0
    c*det(0) = 0
     
  2. jcsd
  3. Oct 21, 2010 #2
    [tex]\begin{pmatrix}1&0\\0&0\end{pmatrix}+\begin{pmatrix}0&0\\0&1\end{pmatrix}=\ldots[/tex]
     
  4. Oct 21, 2010 #3
    Sorry, but can you explain what you meant? Thanks
     
  5. Oct 21, 2010 #4
    Can you add these two matrices? Are they both singular? Is their sum singular? Is the set of singular matrices a vector space?
     
  6. Oct 21, 2010 #5
    They are both singular and if you add them up the result would be a nonsingular matrix. Singular matrices don't have a inverse, so they aren't vector spaces.
     
  7. Oct 21, 2010 #6
    The last sentence is not a good one. In fact it is very very bad (it would be a good exercise for you to find out why it is so bad). A good one is:

    In a vector space, for any two vectors from this space, their sum should be again a vector in the same space.

    The examples show that this is not the case with singular matrices: one can find examples of two singular matrices whose sum is not a singular matrix. Therefore the set of all singular matrices does not satisfy one of the necessary requirements to be a vector space. Therefore it is not a vector space.
     
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