Dimension of Ker(T) when $T(A) = \operatorname{Tr} {(A)}$. For a positive integer $n > 1$, let $ T: \mathbb{R}^{n\times n} \to \mathbb{R}$ be the linear transformation defined by $T(A) = \operatorname{Tr} {(A)}$, where $A$ is an $n \times n$ matrix with real entries. Determine the dimension of $\ker{(T)}$.
I can see just by looking at this transformation that it's not one-to-one. But I'm sort of stuck. I know the zero matrix and matrices with diagonal entries of 0 are examples of tr(A) = 0. But I still can't really see what the dimension is. Any help?

The best way is to use the dimension theorem. What is the dimension of the range of $T$?
