Defining the standard basis vectors

Question

PROBLEMS

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.