Diagonal Matrix Representation : Linear Mapping, Basis and Kernels

A linear mapping T: R3 → R2 is defined on the standard basis vectors via:
T (e1) = (0, 0), T (e2) = (1, 1), T (e3) = (1, -1)

i. Calculate T(4,-1,3)
ii. Find the dimension of the range of T and the dimension of the kernel of T.
iii. Find the matrix representation of T relative to the standard bases in R3, R2.
iv. Find bases {v1, v2, v3} for R3 and {w1, w2} for R2 with respect to which T has diagonal matrix representation.

Please provide a detailed solution to all parts of the above question.

Also, please give a little background theory regarding part 4 or at least a definition for diagonal matrix representation.
For comparison,
my answers to the first three parts are:

i. (2,-4)
ii. dim imT = 2, dim kerT = 1
iii. the matrix below:

0 1 1
0 1 -1