Let Pnx be the space of polynomials in x of degree less than

Let P_n(x) be the space of polynomials in x of degree less than or equal to n, and consider the derivative operator d/dx : P_n(x) rightarrowP_n(x).Find the dimension of the kernel and image of this operator. What happens if the target space is changed to P_n- 1 (x) or P_n + 1(x)?Now consider P_2(x, y), the space of polynomials of degree two or less in x and y. (Recall how degree is counted; xy is degree two. y is degree one and x^2y is degree three, for example.) LetL: =: P_2 (x, y) rightarrow P_2 (x, y)(For example, L(xy) =(xy) +(xy) = y + x.) Find a basis for the kernel of L. Verify the dimension formula in this case.

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