Linear endomorphism of a vector space of matrices

Let M4(R) denote the 16-dimensional real vector space of 4×4 matrices with real entries, in which the vectors are represented as matrices. Let T:M4(R)M4(R) be the linear transformation defined by T(A)=AA.

  1. Determine the dimension of ker(T).
  2. Determine the dimension of im(T).