(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let W1 be the set of all nxn skew-symmetric matrices with entries from a field F. Assume F is not characteristic 2 and let W2 be a subspace of M_{nxn}(F) consisting of all nxn symmetric matrices. Prove the direct sum of W1 and W2 is M_{nxn}(F).

2. Relevant equations

3. The attempt at a solution

I'll just do this for n = 3 for ease of formatting.

Assume A is in W1 and B is in W2.

A = [itex]\left( \begin{array}{ccc}

0 & b & c \\

-b & 0 & d \\

-c & -d & 0 \end{array} \right)[/itex]

B = [itex]\left( \begin{array}{ccc}

e & f & g \\

f & e & h \\

g & h & e \end{array} \right)[/itex]

A+B = [itex]\left( \begin{array}{ccc}

e & b+f & c+g \\

f-b & e & d+h \\

g-c & h-d & e \end{array} \right)[/itex]

For elements a_{x,y}= a_{y,x}in A they must both equal 0. ∴ W1 [itex]\cap[/itex] W2 = {0}

So I have one part of the direct sum proof but the W1+W2 = M_{nxn}(F) part isn't working for me, clearly the diagonal entries can't be all the same.

I guess the first thing I'm wondering about is how to use the assumption that F isn't characteristic 2. I don't see how that assumption helps me at all.

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# Homework Help: Skew-symmetric matrices and subspaces

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