show that: Trace of a matrix squared [(TrA)^2], is a linear function of A tensor A

I have to show that if is a tensor then is a linear function of .

My approach:

If I define a function by (einstein summation is used, and ) is the coefficient respect to a basis, then I have:

and

I'm not quite sure about the identity I have used at the , but maybe it follows from:

Let be a basis such that

and

then

by uniqueness of the coefficients i get:

I think I could proove this in the more general case wich I have used where I use a theorem that states that for any tensor (r,s) there are unique coefficients , but there are so much indices so I don't want to wright it here.