Linear Algebra Subspace Basis Problem

  • #1
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1. The set of all traceless (nxn)-matrices is a subspace sl(n) of (bold)K^(nxn). Find a basis for sl(n). What is the dimension of sl(n)?

Not sure how to go about finding the basis. I know a basis is a list of vectors that is linearly independent and spans.
and for the dimension of sl(n), is it just n^2, as it's an nxn matrix.

any help would be greatly appreciated.
 

Answers and Replies

  • #2
Dick
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A traceless matrix has to satisfy exactly one extra condition relative to a general matrix. What does this tell you about it's dimension? Start by describing n^2-n matrices that have zeros down the diagonal. Then describe the matrices that have only diagonal elements but are still traceless.
 
  • #3
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why is it n^2-n matrices?
I only know that traceless is where the sum of the diagonals is zero.
and I also have the following information from another problem:

the set sl(n) of all traceless (nxn)-matrices form a subspace of (bold)K^(nxn):
the zero matrix belongs to sl(n) so sl(n) is not empty.
tr(cA+B) = tr(cA) + tr(B)
tr(cA+B) = c tr(A) + tr(B)
tr(cA+B) = c (0) + 0 = 0
and cA+B (epsilon) sl(n) which means sl(n) is a subspace.
 
  • #4
Dick
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A basis for the subspace of matrices that have all zeros down the diagonal has dimension n^2-n. Why? Construct an explicit basis. Now think about the matrices that are zero off the diagonal and only have nonzero elements along the diagonal but are traceless. Just sort of 'add the two'.
 
  • #5
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okay, i understand now why the dimension is n^2-n
but do we need the dimension to find the basis. how do i start with the basis, we were given examples in class of our "favorite bases", and i remember the prof saying that we need them to find other bases.
 
  • #6
Dick
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Only the dimension of the off diagonal elements is n^2-n. You have to add some elements to complete the basis for sl(n). They are purely diagonal. How many are there?
 

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