Find the dual basis, as defined in Exercise 7.1.32, for the monomial basis of P (2) with
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Find the dual basis, as defined in Exercise 7.1.32, for the monomial basis of P(2) with respect to the L2 inner product
Data From Exercise 7.1.32
Dual Bases: Given a basis v1, . . . , vn of V, the dual basis ℓ1, . . . , ℓn of V∗ consists of the linear functions uniquely defined by the requirements
(a) Show that ℓi[v] = xi gives the ith coordinate of a vector v = x1v1 + · · · + xnvn with respect to the given basis.
(b) Prove that the dual basis is indeed a basis for the dual vector space.
(c) Prove that if V = Rn and A = (v1 v2 . . . vn) is the n × n matrix whose columns are the basis vectors, then the rows of the inverse matrix A−1 can be identified as the corresponding dual basis of (Rn)∗.
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