Let $S$ be the $ℝ$-vector subspace of the matrices $A ∈ ℂ^{3x3}$ such that the sum of the elements of its diagonal equals zero. Determine a basis and the dimension of that subspace.
I attach this image where G is the set of generators that I found. Anyway I am not sure if it is correct since the being of complex matrices and the subspace is an $ℝ$-vector subspace I find difficult to see which vectors could determine this subspace and form a base.
The condition of having trace equal to (the complex number) $0$ is a nontrivial complex-linear condition, so the solution set is a complex vector space of dimension $9-1=8$. However, your are asked for its dimension as a real vector space, and then the dimension is $2\times8=16$. For a basis of the real vector space, start with a basis as a complex subspace, then add the complex multiple by$~\mathbf i$ of each basis vector.