ts) Let P2 denote the vector space of polynomials of degree up to 2. Which of the following subsets of P2 are subspaces of P2? A. {p(t) | p' (t) + 8p(t) + 4 = 0} B. {p(t) | p(-t) = p(t) for all t} Oc. {p(t) | fp(t)dt = 0} D. {p(t) |p(1)=0} OE. {p(t) | p (2) = p(5)} OF. {p(t) |p(4) = 1}

Find the orthogonal projection of v=(−19,6,18)onto the subspace W of ℝ3 spanned by (−7,6,−2) and (0,6,18) ProjW(v)=

= 4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ = {ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by Define T: P2 P₂ by (p\q) = p(x)q(x) dx. 0 T(ao + ax + a2x²) = a₁x. (a) Show that T is not Hermitian (i.e. self-adjoint). (b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian. Explain why this does not contradict with result from Part (a).

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