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5 whose elements bU are from span {1, x, x2, x3}. ) (b) For R[I] with basis b = (1, x, y, x2)T, find the multiplication matrices Ax and Ay\ cf. 6. 2. , bm)T and the multiplication matrices Aa. Convince yourself that where aJM is the /^th row of Aa. 8, list the polynomials pavi. What happens when a^ is a trivial row? (b) The four nontrivial pa{jL constitute a basis of I, but not a minimal one. 2 to select a minimal basis. (c) For the data of Exercise 1 b) above, find the pafl and select a minimal basis.

For ft[J], with basis b and related multiplication matrices Aa (cf. 6), there is a unique dual space (R[I])*. 27). 16. , cm) be the conjugate basis of the basis b = ( & i , . . , bm)T in R[I]. 15, cTxa = cr EM(W Afc|i) = cr ACT. 31) follows by linear superposition. Now we remember that each element r e R|T] is the representative of the residue class [r] j c Ps. Thus we can extend the domain of the functionals in (R[I])* to all of Ps by letting them take the same value on all polynomials in the same residue class [r]j.

C) For the data of Exercise 1 b) above, find the pafl and select a minimal basis. 3. 8. Normalize the eigenvectors such that their first components are 1. 2. 1 Dual Spaces of Polynomial Ideals Dual Vector Spaces A particularly useful feature of linear spaces or vector spaces is the fact that they have natural dual counterparts: the spaces of all linear functionals on a given vector space form another vector space which mirrors the properties of the original one. This fundamental insight from linear algebra is also important and useful in polynomial algebra.

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