I know there is a result that $S=$ collection of all trace $0$ matrices, and that collection forms a vector space. But I want to prove it independently i.e.
For any $A,B,C,D\in M_n(K)$ we have to find $E,F\in M_n(K)$ such that $(AB-BA)+(CD-DC)=EF-FE$.
But I don't know how to solve this. But the thing I can observe that the above equation gives rise to $n^2$ equation (equating each entries of matrices of both the sides) with $2n^2$ variables (Total number of entries of $E,F$ is $2n^2$).Can this problem be simplified if we choose special kind of $E$ say diagonal matrix.
Edit-(This is valid only for $\Bbb{R}$ or $\Bbb{C}$)
$AB-BA=(A+aI)(B+bI)-(B+bI)(A+aI)$ for all $a,b\in\Bbb{R}$. And there is $a,b$ in $\Bbb{R}$ such that $A+aI,B+bI$ are invertible.
Hence, we can assume $A,B$ to be invertible matrices i.e. $S=\{AB-BA|A,B\in GL_n(K)\}$
from Hot Weekly Questions - Mathematics Stack Exchange
Biswarup Saha
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