I understand what the basis must look like by considering lower dimensions, but I don't know how to represent the basis in a set.
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"I understand what the basis must look like by considering lower dimensions": what do you mean by that? Can you come up with a basis, for example, of the $2 \times 2$ or $3 \times 3$ skew-symmetric matrices? – Ben Grossmann Sep 30 '15 at 16:10
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What is a lower dimension? – copper.hat Sep 30 '15 at 16:11
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ya that's what i meant – bean4 Sep 30 '15 at 16:11
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Do you know what a basis is? – copper.hat Sep 30 '15 at 16:11
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yup, it's a set that needs to span the subspace and it needs to be linearly independent – bean4 Sep 30 '15 at 16:12
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Do you see any relationship between the set of skew matrices and the set of strictly upper triangular matrices? Can you find a basis for the latter, and if so, how to modify it so that it forms a basis for the former? – copper.hat Sep 30 '15 at 16:14
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Hint: For a given $n$, let $E_{ij}$ denote the $n \times n$ matrix whose entries are all zero except for the $i,j$ entry, which is a $1$. Consider the matrices $E_{ij} - E_{ji}$. How can we use these to form a basis?
Ben Grossmann
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Hints:
- Skew-symmetric matrices are specified by the part above the diagonal (why?).
- What is a basis for these upper-triangular matrices?
- How do I decompose a matrix into symmetric and skew-symmetric parts?
Chappers
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