# Thread: Do they form a vector space?

1. ## Do they form a vector space?

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.

2. 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?

3. Originally Posted by HallsofIvy
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?
Yes, I think it is true that TR(c*A)= c*Tr(A) when c is any complex number. Am I right? And if so, does it mean that the basis and the dimension of the vectorspace are the same over the field R and over the field C?
Thanks for helping me!