(Solution) - Let T P2 R3 be defined by T p p 0 -(2025 Original AI-Free Solution)

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Let T: P2 ? R3 be defined by
T(p) = (p(0)), p(1),(p(2)) for all p in P2. Let
B = {1,x, x2}and D = [(1,0,0),(0,1,0),(0,0,1)]
(a) show that MDB (T) =

And conclude that T is an isomorphism.
(b) Generalize to T: Pn ? Rn+1 where
T(P) = (P(a0) p(a1..., p(an)) and a0, au...,a" are distinct real numbers.
[Hint: Theorem 7 §3.2.