For p(t) = a2t2 + a1t + a0 and r(t) = b2t2 + b1t + b0 in P2[t], define<p(t), r(t)> := a2b2 + a1b1 + 3a0b0.a.) Show that <p(t), r(t)> as defined above is an inner product on P2[t].b.) If p(t) = t2 − 2, r(t) = t + 1 and s(t) = 3t2 − t, compute for kp(t)k and for d(r(t), s(t)).c.) Let S = {t2 − 2, t + 1}. Using the given inner product defined above,compute for a basis of S⊥.

a.<p(t),r(t)>:=a2b2+a1b1+a0b0<p(t),r(t)> is an inner product if it satisfies three…

Answered: For p(t) = a2t2 + a1t + a0 and r(t) =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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For p(t) = a₂t² + a₁t + a₀ and r(t) = b₂t² + b₁t + b₀ in P₂[t], define
<p(t), r(t)> := a₂b₂ + a₁b₁ + 3a₀b₀.

a.) Show that <p(t), r(t)> as defined above is an inner product on P₂[t].
b.) If p(t) = t² − 2, r(t) = t + 1 and s(t) = 3t² − t, compute for kp(t)k and for d(r(t), s(t)).
c.) Let S = {t² − 2, t + 1}. Using the given inner product defined above,
compute for a basis of S^⊥.

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