CHAPTER 4 Matrices and Linear Systems of Equations_v2

CHAPTER 4 Matrices and Linear Systems of Equations_v2 - 1...

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Unformatted text preview: 1 Chapter 4 Matrices and Systems of Linear Equations Contents Introduction ......................................................................................................................... 1 Background ......................................................................................................................... 2 More Problems with Round-Off Error ........................................................................... 4 4.1 MATRICES ................................................................................................................. 4 MATRIX ALGEBRA ....................................................................................................... 7 MATRIX ADDITION ...................................................................................................... 8 CRAMERS RULE .......................................................................................................... 17 4.2 Ill-Conditioned Matrices, Round-Off Error, and Classification of Linear Systems of Equations ...................................................................................................... 33 VISUALIZING CASES #1, #2, #3, and #4 ................................................................... 35 Example of an Ill Conditioned System of Equations: The Hilbert Matrix ................ 42 CHAPTER 4.1 MATRICES SUMMARY .................................................................... 52 CHAPTER 4.2 Ill-Conditioned Matrices and Classification of Linear Systems of Equations ......................................................................................................................... 53 Exercises ........................................................................................................................... 54 Introduction Systems of linear equations are enormously important in science and engineering. Routine applications range from mass balances in a process for making gasoline or design of simple electrical circuits, to modeling sales in a retail chain or predicting the lengths of lines in an amusement park. Some applications may result in thousands of equations and thousands of unknowns. One source of these large systems of equations is use of a grid over the region of definition of a system of partial differential equations. This changes the problem of solving a system of partial differential equations into the problem of solving a large, linear system of equations. These are among the most powerful applications of numerical methods and are at the leading edge of disciplines from meteorology to hydrology to aerospace engineering. Background If you passed algebra you are approach by which we solve is a linear equation: y = mx + straight line that we have alre m is the slope and b is the int The simplest system of linear Equations (4.1) are a system that when we first write the tw simultaneously satisfy (solve y = 4x - 2 1 y = x + 4 Figure 4.1 If we plot these two equation one point where there is a sol...
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This note was uploaded on 01/26/2011 for the course CH E 2112 taught by Professor Dr.harwell during the Spring '10 term at The University of Oklahoma.

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CHAPTER 4 Matrices and Linear Systems of Equations_v2 - 1...

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