Hello all! I finally decided to join this forum after quite a long time of lurking haha. Anywho, to the point. A friend of mine is taking linear algebra and is having a lot of issues with the below posted problem. I'm trying to help her with it, but sadly I'm having issues with it as well haha. I would appreciate any help you all could offer me in doing this proof. I've gotten it all solved up until I have to prove that [itex]M_{n\times n}(F) = W_1 \oplus W_2[/itex]. So basically, I'm on the last step.

Prove:

A matrix M is called a skew-symmetric if [itex]M^t = -M[/itex]. Clearly, a skew-symmetric matrix is square. Let F be a field. Prove that the set [itex]W_1[/itex] of all skew-symmetric n x n matrices with entries from F is a Subspace of [itex]M_{n \times n}(F)[/itex]. Now assume that F is not of characteristic 2, and let [itex]W_2[/itex] be the subspace of [itex]M_{n \times n}(F)[/itex] conisting of all symmetric n x n matrices. Prove that [itex]M_{n\times n}(F) = W_1 \oplus W_2[/itex]

Prove:

A matrix M is called a skew-symmetric if [itex]M^t = -M[/itex]. Clearly, a skew-symmetric matrix is square. Let F be a field. Prove that the set [itex]W_1[/itex] of all skew-symmetric n x n matrices with entries from F is a Subspace of [itex]M_{n \times n}(F)[/itex]. Now assume that F is not of characteristic 2, and let [itex]W_2[/itex] be the subspace of [itex]M_{n \times n}(F)[/itex] conisting of all symmetric n x n matrices. Prove that [itex]M_{n\times n}(F) = W_1 \oplus W_2[/itex]

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