Basis for Kernel and Trace of Square Matrices

Question

I'm supposed to find the basis for the kernel and the image of $T$, and I know that $$T(A) = Tr(A)=\sum^n_{k=1}a_{kk}=a_{11}+\dots+a_{nn},$$ but to me it just look like scalar. I know that bases are supposed to consist of vectors that span whatever space or linear transformation your in, so would the basis $\beta_1$ for the kernel of $T$ be the entire map? And, what about the basis $\beta_2$ for the image of $T$? Thanks in advance.