Let T: M_{2x2}(R)------>P_{3}(R) be a linear transformation defined by

T ([a b]) = (a-b) + (a-d)x +(b-c)x^{2}+(c-d)x^{3}

[c d]

Consider the bases v ={[1 0] , [0 1] ,[1 0] ,[ 0 0]} of M_{2x2}

- - - -------- -- -- ---- [1 0 ] [0 1] [0 1] [1 1]

and u = {x, x-x^{2},x-x^{3},x-1} of P_{3}(R)

a) find [T]_{uv }

b) use [T]_{uv}to find a basis for the kernel of T

c) use [T]_{uv}to find a basis for the image of T

d) State the nullity and rank of T. Is T injective? surjective?