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Lecture_03_Matrix_Math___Gauss_Eliminati

# Lecture_03_Matrix_Math___Gauss_Eliminati - EGR 102...

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EGR 102 Lecture 3 1 EGR 102 Introduction to Engineering Modeling Matrix Math Chapter 8.1 Gauss Elimination Chapter 9.1.1, 9.1.3, 9.2 & 9.3 Figures from: “Applied Numerical Methods with MATLAB,” Steven Chapra, McGraw Hill EGR 102 Lecture 3 2 MATLAB: Matrix Laboratory Images, movies, sound and variables all are matrices in MATLAB Sound: load chirp sound(y) Image: load clown image(X), colormap(map) Animations (movies): z = peaks; surf(z); axis tight set(gca,'nextplot','replacechildren'); for j = 1:20 surf(sin(2*pi*j/20)*z,z) f(j) = getframe; end movie(f,50)

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EGR 102 Lecture 3 3 MATLAB MATLAB is both a software tool and a MATLAB is both a software tool and a computer language. A variety of numerical functions, symbolic computations, and visualization tools have been added to the matrix manipulations. Other languages & libraries include: FORTRAN (IMSL), C++, FORTRAN (IMSL), C++,… EGR 102 Lecture 3 4 Numerical Methods … Solving a system of algebraic equations / inverting matrices Finding roots of equations Solving matrices (linear algebra) Regression / Curve-fitting Numerical Integration Solving ODE’s (Ordinary Differential Equations) More advanced topics ( e.g., solving PDEs) …
EGR 102 Lecture 3 5 Finite Element Analysis Millions of simultaneous equations EGR 102 Lecture 3 6 Matrix Math Review Matrices used to: Describe linear equations Track coefficients of linear transformations Record data that depend on multiple parameters Matrices can be added, multiplied, and decomposed in various ways

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EGR 102 Lecture 3 7 Matrix Math Review The horizontal lines in a matrix are called rows The vertical lines are called columns . A matrix with m rows & n columns is called an m -by- n matrix m and n are called its dimensions . The dimensions of a matrix are always given with the number of rows first, then the number of columns. It is commonly said that an m -by- n matrix has an order (size) of m × n . = 12 11 10 9 8 7 6 5 4 3 2 1 A EGR 102 Lecture 3 8 Matrix Math Review Almost always capital letters denote matrices with the corresponding lower-case letters with two indices representing the entries. For example, the entry of a matrix A that lies in the i -th row and the j -th column is written as a i,j and called the i,j entry or (i,j) -th entry of A . Alternative notations for that entry are A [ i,j ] or A i,j . The row is always noted first, then the column. = 12 11 10 9 8 7 6 5 4 3 2 1 A a 2,3 = ?