By Ron Larson
The cornerstone of hassle-free LINEAR ALGEBRA is the authors' transparent, cautious, and concise presentation of material--written in order that readers can absolutely know the way arithmetic works. This application balances idea with examples, purposes, and geometric instinct for a whole, step by step studying process. that includes a brand new layout that highlights the relevance of the math and improves clarity, the 7th variation additionally contains new conceptual Capstone workouts that strengthen a number of ideas in every one part. facts and functions replicate present statistics and examples to interact clients and reveal the hyperlink among conception and perform
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Extra resources for Elementary linear algebra
3 22. Generalizing the statement in Exercise 21, if a polynomial function p͑x͒ ϭ a0 ϩ a1x ϩ . . ϩ anϪ1x nϪ1 33 Exercises 28. Network Analysis The figure shows the flow of traffic (in vehicles per hour) through a network of streets. x1 300 is zero for more than n Ϫ 1 x-values, then 150 a0 ϭ a1 ϭ . . ϭ anϪ1 ϭ 0. Use this result to prove that there is at most one polynomial function of degree n Ϫ 1 (or less) whose graph passes through n points in the plane with distinct x-coordinates. 23. Calculus The graph of a cubic polynomial function has horizontal tangents at ͑1, Ϫ2͒ and ͑Ϫ1, 2͒.
Y (xn, yn) (x3, y3) POLYNOMIAL CURVE FITTING Suppose n points in the xy-plane ͑x 1, y1͒, ͑x 2 , y2 ͒, . . , ͑x n , yn ͒ represent a collection of data and you are asked to find a polynomial function of degree nϪ1 (x2, y2) x (x1, y1) p͑x͒ ϭ a0 ϩ a1x ϩ a2 x 2 ϩ . . ϩ anϪ1 x nϪ1 whose graph passes through the specified points. This procedure is called polynomial curve fitting. 4. To solve for the n coefficients of p͑x͒, substitute each of the n points into the polynomial function and obtain n linear equations in n variables a0 , a1, a 2 , .
The matrix containing only the coefficients of the system is called the coefficient matrix of the system. Here is an example. System x Ϫ 4y ϩ 3z ϭ 5 Ϫx ϩ 3y Ϫ z ϭ Ϫ3 2x Ϫ 4z ϭ 6 Augmented Matrix ΄ 1 Ϫ1 2 Ϫ4 3 0 3 Ϫ1 Ϫ4 5 Ϫ3 6 Coefficient Matrix ΅ ΄ 1 Ϫ1 2 Ϫ4 3 0 3 Ϫ1 Ϫ4 ΅ 14 Chapter 1 Systems of Linear Equations ELEMENTARY ROW OPERATIONS In the previous section, you studied three operations that produce equivalent systems of linear equations. 1. Interchange two equations. 2. Multiply an equation by a nonzero constant.