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Given the subspace U{a(t+1)^2 + b | a,b in R} and W={a+bt+(a+b)t^2+(a-b)t^3 |a,b in R}

1) show that V = U direct sum with W

2) Find a basis for U perp for some inner product

Attempt at the solution:

1) For the direct sum I need to show that it satisfy two conditions (a) V = U + W (which I did), (b) the intersection of U and W = {0} which I am not quite sure how to do.

2) expanding U: a(t^2 + 2t +1) +b. as far as I understand U perp needs only to contain t^3 so that U + U perp span P3 (right?).

Next I used the standard inner product for polynomials such that the integral from 0 to 1 of U.U perp = 0.

so I have the following (at^2 + 2at +a +b)(-bt^3) integrate from 0 to 1, set it equal to zero and solve for b in terms of a. (here I chose U perp to be -bt^3 from some b in R).

Is solution valid for finding a basis for U perp? Thanks