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Dimension of a subspace

  1. Mar 28, 2012 #1
    Find the dimension of the subspace of M2;2 consisting of all 2 by 2 matrices
    whose diagonal entries are zero. ?

    i know that the dimension is the number of vectors that are the basis for this subspace ,but i cannot figure out what is the basis for this subspace ?

    any help will be appreciated ,
    thanks in advance
     
  2. jcsd
  3. Mar 28, 2012 #2

    Bacle2

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    Try doing a generalized row-reduction using Gaussian elimination, to see how many parameters determine the subspace.
     
  4. Mar 28, 2012 #3
    Question:

    What's the difference between a 2 by 2 matrix and a column vector with 4 entries?

    Presumably, you know how to find a basis for all the column vectors with 4 entries.
     
  5. Mar 28, 2012 #4
    so , Such a matrix is of the form
    [0 b]
    [c 0] for some b, c.

    Since these matrices are generated by
    [0 1].[0 0]
    [0 0],[1 0], the dimension equals 2. is that a right answer ?
     
  6. Mar 28, 2012 #5
    Yes, that's it.
     
  7. Mar 29, 2012 #6

    Bacle2

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    I don't mean to use rigor, for rigor's sake, but I think it is important to note that not only
    do those matrices generate, but that no smaller ( in size/cardinality) set generates the whole space.
     
  8. Mar 29, 2012 #7

    HallsofIvy

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    Specifically, any matrix in that set can be written
    [tex]\begin{bmatrix}0 & a \\ b & 0\end{bmatrix}= \begin{bmatrix}0 & a \\ 0 & 0 \end{bmatrix}+ \begin{bmatrix}0 & 0 \\ b & 0 \end{bmatrix}= a\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}+ b\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}[/tex]
    which makes it clear what a basis is.
     
  9. Mar 31, 2012 #8
    WHAT IS THE RESULT span(spanV)=???
     
  10. Mar 31, 2012 #9
    Try doing a generalized row-reduction using Gaussian elimination.
     
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