Linear Algebra: Polynomials subspaces

[j.ayoub]

U and W are subspaces of V = P3(R)
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

 

[Mark44]

For the b part of 1, you might try this.
U = {a(t + 1)² + b | a, b in R}
W = {(c - d)t³ + (c + d)t² + dt + c | c, d in R}
(Notice that I changed to c and d in the 2nd subspace.)

You can split these up like so:
U = { a(t² + 2t + 1) + b(1)}
W = {c(t³ + t² + 1) + d(-t³ + t² + t)}

If you think about the generators of each subspace as vectors (which are probably a bit easier to work with), U is all linear combinations of <0, 1, 2, 1> and <0, 0, 0, 1>, and W is all linear combinations of <1, 1, 0, 1> and <-1, 1, 1, 0>.

If you can show that these vectors are linearly independent, that means that none of them is a linear combination of the others, and that means that the intersection of the two subspaces is {0}.

 

[j.ayoub]

Thanks Mark44 for your help