Greetings , got stuck on the following queston

Let M(3x3) be vector space of 3x3 matrices , let F : M(3x3) ---> M(3x3) be a linear transformation defined the following way :

for every A in M(3x3) be F(A) = B is a symmetrice matrice

F(A) = 1/2(A+A(T)) -1/3*Tr(A)*I

Tr(B) = 0

Find a basis for Ker F and Im F

anyways while playing around with Ker F i found the way the matrix must look like for the Ker F

also i think i got a basis for the ker F , but not sure how to go about with finding the Im F , also is my basis right ? here it is