Linear Least Squares Suppose we are given a set of data points {(xi ‚ fi )}‚ i = 1‚ . . . ‚ n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. You have seen that we can interpolate these points‚ i.e.‚ either find a polynomial of degree ≤ (n − 1) which passes through all n points or we can use a continuous piecewise interpolant of the data which is usually a better approach. How‚ it might be the case that we know that these data points should
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.. 15 Matrices .................................................................................................................................................... 27 Matrix Arithmetic & Operations .............................................................................................................. 33 Properties of Matrix Arithmetic and the Transpose ................................................................................. 45 Inverse Matrices and Elementary Matrices ............
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1.1 SOLUTIONS Notes: The key exercises are 7 (or 11 or 12)‚ 19–22‚ and 25. For brevity‚ the symbols R1‚ R2‚…‚ stand for row 1 (or equation 1)‚ row 2 (or equation 2)‚ and so on. Additional notes are at the end of the section. 1. x1 + 5 x2 = 7 −2 x1 − 7 x2 = −5 1 −2 5 −7 7 −5 x1 + 5 x2 = 7 Replace R2 by R2 + (2)R1 and obtain: 3x2 = 9 x1 + 5 x2 = 7 x2 = 3 x1 1 0 1 0 1 0 5 3 5 1 0 1 7 9 7 3 −8 3 Scale R2 by 1/3: Replace R1 by R1
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The African Americans were brought from Africa in a system called the Triangular Trade. The triangular trade lasted from 1450 to 1750. Not long after the trade ended‚ America had divided into two parts‚ the Union and the Confederacy. Over the course of 4 years‚ the bloodiest war fought in United States history broke out‚ tearing America apart. The North and South fought gruelingly through the seasons‚ both determined for a strong win. The African Americans gained freedom on May 9‚ 1865. The African
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Math 111 Homework 1 fall 2007 due 14/9 1. (1.2; 17) Determine the values of h such that the matrix is the augmented matrix of a system which admits a solution. 2 3 4 6 h 7 2. (1.2; 12) Find the general solutions of the system whose augmented matrix is 1 −7 0 6 5 0 0 1 −2 −3 −1 7 −4 2 7 1 −2 4 3. (1.3; 17) Let a1 = 4 ‚ a2 = −3 ‚ b = 1 . For what −2 7 h value(s) of h is b in the plane spanned by a1 and a2? 4. (1.4; 15) Let A = b1 2 −1 and b = . Show that the equation
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trade changed the Atlantic Ocean from a predominantly unclaimed vast ocean into part of the growing maritime empires‚ booming with trade. As the region progressed‚ economic‚ political‚ and social changes occurred rapidly due to the emergence of the Triangular Trade Route and the Trans- Atlantic Slave Trade. By the late 1600s‚ the increased participation in these trade routes allowed a multitude of commodities to reach the Atlantic World‚ permitting Europeans to construct big maritime empires and constantly
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A SAMPLE RESEARCH PAPER/THESIS/DISSERTATION ON ASPECTS OF ELEMENTARY LINEARY ALGEBRA by James Smith B.S.‚ Southern Illinois University‚ 2010 A Research Paper/Thesis/Dissertation Submitted in Partial Fulfillment of the Requirements for the Master of Science Degree Department of Mathematics in the Graduate School Southern Illinois University Carbondale July‚ 2006 (Please replace Name and Year with your information and delete all instructions) Copyright by NAME‚ YEAR All Rights Reserved
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Examples in Plotting 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Matrices—Two–Dimensional Arrays 13 16.1 Size of a matrix . . . . . . . . . . . . 14 16.2 Transpose of a matrix . . . . . . . . 14 16.3 Special Matrices . . . . . . . . . . . 14 16.4 The Identity Matrix . . . . . . . . . 14 16.5 Diagonal Matrices . . . . . . . . . . 15 16.6 Building Matrices . . . . . . . . . . . 15 16.7 Tabulating Functions . . . . . . . . . 15 16.8 Extracting
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number of dots. Finally‚ we can generate a simplified formula or general statement for to find the number of dots in each stage. Triangular numbers are sequence of number that has certain pattern and if we plot it into a diagram it will create a geometrical triangular shape. Each triangular number has its own number of dots and formula. Now‚ we will start with triangular number first. Aim: In this task I will consider geometric shapes which lead to special numbers. The simplest example of these
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square numbers‚ 1‚ 4‚ 9‚ 16‚ which can be represented by squares of side 1‚ 2‚ 3 and 4. The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers (1‚3‚6‚)…. 1 3 6 10 15 Complete the triangular numbers sequence with three more terms. Find a general statement that represents the nth triangular number in terms of n. Consider stellar (star) shapes with p vertices‚ leading to p-stellar numbers. The first four representations
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