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I'm currently studying by the book "Theory of Lie groups", C. Chevalley. On page 6, paragraph before Proposition 6, he says: "The sets $M^S$, $M^{sh}$, $M^R$, $M^S$\cap$M^{sh}$, $M^R$\cap$M^S$, $M^R$\cap$M^{sh}$, $M^R$\cap$M^S$\cap$M^{sh}$, $M^s$ may all be considered as vector spaces over the field $R$ of real numbers; as such, their dimensions are $2n^2 - 2$, $n^2$, $n^2$, $n^2 - 1$, $n^2 - 1$, $n^2 - 1$, $n(n-1)/2$, $n(n-1)/2$ and $n(n-1)$ respectively."

$M^S$ is the set of matrices of trace O. $M^{sh}$ is the set of skew-hermitian matrices. $M^R$ is the set of real matrices. $M^s$ is the set of symmetric matrices.

I didn't understand how he concluded that each subset has the respective dimension he says.

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    Sorry, I've put three "n²-1". Actually, there are just two. I didn't understand it too, but this is how it is written =/2012-08-19

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