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Since (1, x, x2 , . . , xn ) is a basis for V , clearly dim(V ) = n + 1. Hence B = (p0 , . . , pn ) must be a basis for V . It then follows from Eq. 6 that for each f ∈ V , we have n f= f (ti )pi . 7) i=0 The expression in Eq. 7 is known as Lagrange’s Interpolation Formula. Setting f = xj in Eq. 8) i=0 Definition Let A1 and A2 be two linear algebras over F . 3. LAGRANGE INTERPOLATION 53 and (b) (uv) = u v for all u, v ∈ A1 and all scalars c, d ∈ F . The mapping u → u is called an isomorphism of A1 onto A2 .

For example 6 3 = 20 ≡ 2 (mod 3), even though 3! is zero modulo 3. The derivative of the polynomial f = c0 + c1 x + · · · + cn x n is the polynomial f = Df = c1 + 2c2 x + · · · + ncn xn−1 . Note: D is a linear operator on F [x]. 2. (Taylor’s Formula). Let F be any field, let n be any positive integer, and let f ∈ F have degree m ≤ n. Then n f= k=0 Dk (f )(c) (x − c)k . k! Be sure to explain how to deal with the case where k! is divisible by the characteristic of the field F . If f ∈ F [x] and c ∈ F , the multiplicity of c as a root of f is the largest positive integer r such that (x − c)r divides f .

In this chapter, which we usually assign as independent reading, we wish to give the student a somewhat formal introduction to the algebra of polynomials over a field. It is then natural to generalize to polynomials with coefficients from some more general algebraic structure, such as a commutative ring. The title of the course for which this book is intended includes the words “linear algebra,” so we feel some obligation to define what a linear algebra is. Definition Let F be a field. A linear algebra over the field F is a vector space A over F with an additional operation called multiplication of vectors which associates with each pair of vectors u, v ∈ A a vector uv in A called the product of u and v in such a way that (a) multiplication is associative: u(vw) = (uv)w for all u, v, w ∈ A; (b) multiplication distributes over addition: u(v + w) = (uv) + (uw) and (u + v)w = (uw) + (vw), for all u, v, w ∈ A; (c) for each scalar c ∈ F , c(uv) = (cu)v = u(cv) for all u, v ∈ A.

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

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