Let V - P(1) be the vector space consisting of linear polynomials p(t) = at + b.
(a) Carefully explain why

defines an inner product on V.
(b) Find all polynomials p(t) = a t + b ˆŠ V that are orthogonal to p1(t) = 1 based on this inner product.
(c) Use part (b) to construct an orthonormal basis of V for this inner product.
(d) Find an orthonormal basis of the space P(2) of quadratic polynomials for the same inner product.

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