Applied Engineering Analysis
                               - slides for class teaching*
                                                Chapter 4
                          Linear Algebra and Matrices
                      * Based on the book of  “Applied Engineering  
                         Analysis”, by Tai-Ran Hsu, published by
                         John Wiley & Sons, 2018. 
                          (ISBN 9781119071204)
           (Chapter 4 Linear Algebra and Matrices)                                                1
           © Tai-Ran Hsu
       Chapter Learning Objectives
       • Linear algebra and its applications
       • Forms of linear functions and linear equations
       • Expression of simultaneous linear equations in matrix forms
       • Distinction between matrices and determinants
       • Different forms of matrices for different applications
       • Transposition of matrices
       • Addition, subtraction and multiplication of matrices
       • Inversion of matrices
       • Solution of simultaneous equations using matrix inversion method
       • Solution of large numbers of simultaneous equations using Gaussian elimination method
       • Eigenvalues and Eigenfunctions in engineering analysis                                   2
           4.1 Introduction to Linear Algebra and Matrices
               Linear algebra is concerned mainly with:
                         Systems of linear equations, 
                         Matrices, 
                         Vector space, 
                         Linear transformations, 
                         Eigenvalues, and eigenvectors. 
              Linear and Non-linear Functions and Equations:
                                                                     Linear equations:
                  Linear                                            -4x + 3x –2x + x = 0
                  functions:                                           1     2    3    4
                                                                    where x , x , x and x are 
                                                                            1  2  3      4
                                                                    unknown quantities
               Examples of Nonlinear Equations:                  Simultaneous linear equations:
                             2   3                                  84xx     x12
             4x1 3x2 2x3 x4  0                                   123
                                                                      26xx    x3
             or    x2 + y2 = 1                                          123
                                                                       xx22x
             or    xy = 1                                               123
              or   sinx = y                                       where x1, x2 and x3 are unknown
                                                                  quantities
                                                                                                   3
        4.2 Determinants and Matrices
          Both determinants and matrices are logical and convenient representations of large sets of 
          real numbers or variables and vectors involved in engineering analyses. 
          These large sets of real numbers, variables and vector quantities are arranged in arrays of 
          rows and columns: 
                           a   a   a                  a
                             11  12 13                     1n
                           a21 a22 a23                a2n
                           a31 a32 a33                a3n
                                                    
                                                    
                           am1 am2 am3                amn
          in which a , a ,………………….., a     represent group of data, with m=row number,
                   11 12                mn
          n = column number,  and m = 1,2,3,….,m and n = 1,2,3,…..,n
                                                                                    4