The only difference between "over the field R" and "over the field C" is that, in the second, the scalar, "c", you multiply by could be a complex number. Is TR(c*A)= c*Tr(A) when c is any complex number?
Do n-by-n traceless matrices form a vector space over the field C of complex numbers? Why or why not? If yes, show a basis and give the dimension of the vector space!
Over the field R of real numbers, I know that they form a vector space:
Tr(A)+Tr(B)=Tr(A)+Tr(B)
TR(c*A)=c*Tr(A)
0 matrix is in the vector space
-A matrix is in
But I don't know what is a basis over real numbers.
And I don't even know if they form a vector space over the field C of complex numbers. I suppose they do, but I am not sure, I can't prove it.
Your help would be appreciated.