I have the following problem: for $n>1$ $$V=M_n(\mathbb{K})$$ $$V_1 = \{ A\in V\mid A = A^{\top} \}$$ $$V_2 = \{A\in V\mid A= -A^{\top}\}$$
I need to show that $V_1$ and $V_2$ are subspaces for $V$ and to find a basis for them.
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And here is the basis for $V_2$. – Dietrich Burde Jan 08 '16 at 16:26
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A basis for $V_1$ is given here, which is itself a duplicate. – Dietrich Burde Jan 08 '16 at 16:28
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Hint:
As noted by @ncmathsadist you can easily see that the transpose operation is linear, so $V_1$ and $V_2$ are subspaces.
To find the basis you can start from the case $n=2$ and you easily see that a matrix in $V_1$ can be expressed as: $$ \begin{bmatrix} a&b\\ b&a \end{bmatrix}= a\begin{bmatrix} 1&0\\ 0&0 \end{bmatrix} +b\begin{bmatrix} 0&1\\ 1&0 \end{bmatrix} +c\begin{bmatrix} 0&0\\ 0&1 \end{bmatrix} $$ and, since the matrices at the right are linearly independent, these are a basis.
Can you generalize to $n>2$? and do the same for $V_2$? Starting from the fact that, for $n=2$ a matrix in $V_2$ has the form $$ \begin{bmatrix} 0&b\\ -b&0 \end{bmatrix} $$
Emilio Novati
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