Dimension of a subspace

  • Thread starter baher
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  • #1
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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
 

Answers and Replies

  • #2
Bacle2
Science Advisor
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Try doing a generalized row-reduction using Gaussian elimination, to see how many parameters determine the subspace.
 
  • #3
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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.
 
  • #4
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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
Bacle2
Science Advisor
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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.
 
  • #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.
 
  • #8
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WHAT IS THE RESULT span(spanV)=???
 
  • #9
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Try doing a generalized row-reduction using Gaussian elimination.
 

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