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Catchfire
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Homework Statement
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 Mnxn(F) consisting of all nxn symmetric matrices. Prove the direct sum of W1 and W2 is Mnxn(F).
Homework Equations
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 ax,y = ay,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 = Mnxn(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.