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a) Find the matrix of T with respect to the standard basis B={1,x,x^2} for P2.

T(1) = (x+1) * 0 - 1 = -1 = -1 + 0x + 0x^2

T(x) = (x+1) * 1 - x = 1 = 1 + 0x + 0x^2

T(x^2) = (x+1) * 2x - x^2 = 2x + x^2 = 0 + 2x + x^2

So, the matrix for T with respect to B equals

[-1 1 0]

[0 0 2]

[0 0 1].

b) Find a basis for kerT and hence write down dim(kerT).

c) Find a basis for ImT and hence write down dim(ImT).

d) Does the transformation have an inverse?

I've done part a, so any guidance on the rest would be greatly appreciated.