First prove that dim(Span(3x, 2 + 3/2x^2,4x+x^3)) = 3

Its actually simple. Observe that Span(3x, 2 + 3/2x^2,4x+x^3) has a dimension of 3. Observe that the rank is 3.

maps to . Thus matrix of Transformation is

Its easy to check that the rank of the above matrix is 3.

Hence dimension(image) = rank = 3.

Thus by rank-nullity theorem nullity = 1 and thus the kernel has a dimension 1. This also implies that the Transform is not one-one.