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By S. E. Payne

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Show that f = p 1 f 1 + p2 f 2 + · · · + pk f k . 9. Let f ∈ F [x] have derivative f . Then f is a product of distinct irreducible polynomials over F if and only if f and f are relatively prime. 10. Euclidean algorithm for polynomials Let f (x) and g(x) be polynomials over F for which deg(f (x)) ≥ deg(g(x)) ≥ 1. Use the division algorithm for polynomials to compute polynomials qi (x) and ri (x) as follows. f (x) g(x) r1 (x) r2 (x) = = = = .. rj (x) = rj (x) = q1 (x)g(x) + r1 (x), degr1 (x) < deg(g(x)).

Now we turn our attention to the construction of an algebra which is quite different from the two just given. Let F be a field and let S be the set of all nonnegative integers. We have seen that the set of all functions from S into F is a vector space which we now denote by F ∞ . , lists) f = (f0 , f1 , f2 , . ) of scalars fi ∈ F . If g = (g0 , g1 , g2 , . ) and a, b ∈ F , then af + bg is the infinite list given by af + bg = (af0 + bg0 , af1 + bg1 , . 1) We define a product in F ∞ by associating with each pair (f, g) of vectors in F ∞ the vector f g which is given by n (f g)n = fi gn−i , n = 0, 1, 2, .

Let u be any vector of U , so u = ni=1 ai ui for unique scalars ai ∈ F . Then the desired T has to be defined by T (u) = ni=1 ai T (ui ) = ni=1 ai vi . This clearly defines T uniquely. The fact that T is linear follows easily from the basic properties of vector spaces. Put L(U, V ) = {T : U → V : T is linear}. The interesting fact here is that L(U, V ) is again a vector space in its own right. Vector addition S + T is defined for S, T ∈ L(U, V ) by: (S + T )(u) = S(u) + T (u) for all u ∈ U . Scalar multiplication is defined for a ∈ F and T ∈ L(U, V ) by (aT )(u) = a(T (u)) for all u ∈ U .

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A Second Semester of Linear Algebra by S. E. Payne

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