MATLAB Lecture 2 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr MATLAB Lecture2-- Solving Linear Systems of equations 线性方程组求解 Ref: matlab-Mathematics-Matrices and Linear algebra Solving linear Systems of equations ● Vocabulary: coefficient matrix系数矩阵 linear Systems of Equations线性方程组 row elementary transpositions行初等变换 basis基 backslash反斜线符号 最小二乘解 nonsingular matriⅸx非奇异阵,可逆矩阵 particular solution特解 homogeneous system导出组 olution space解空间 near homogeneous equation线性齐次方程,齐次一次方程 non- homogeneous system非齐次系统(线性方程组) linearly independent线性无关 pseudoinverse广义逆 rational有理数 component元素 determinant行列式 rank秩 overdetermined system超定系统(即方程组无精确解) underdetermined system不定系统(即方程组有无穷多解) orthonormal basis正交基 nul零空间,核空间 Some operations and functions nk det iny rref null o Application on solving linear systems of equations Ax=b, A is an m X n matrix ☆ Review b=0 Theorv r(A)=n, there is only one exact solution zero r(A)<n, there are infinite nonzero solutions r(a)=n, only zero is its exact solution r(A)<n, row elementary transpositions-basis for the null space of A The r(A)=r(a bn, there is only one exact solution r(Ar(a bkn, the ere are infinite solutions r(A)≠r(Ab) there is no exact solution m=n& r(AFr(A b=n, Cramer'rule, A b; row elementary transposition m*n& r(A)=r(A bkn basis for the null space of A, particular solutionMATLAB Lecture 2  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec2­1 MATLAB Lecture2 ­­ Solving Linear Systems of Equations 线性方程组求解 Ref: MATLAB→Mathematics→Matrices and Linear Algebra  →Solving Linear Systems of Equations l Vocabulary: coefficient matrix  系数矩阵 linear Systems of Equations 线性方程组 row elementary transpositions  行初等变换 basis  基 backslash  反斜线符号 least squares solution  最小二乘解 nonsingular matrix  非奇异阵，可逆矩阵 particular solution  特解 homogeneous system  导出组 solution space 解空间 linear homogeneous equation  线性齐次方程，齐次一次方程 non­homogeneous system  非齐次系统（线性方程组） linearly independent  线性无关 pseudoinverse 广义逆 rational  有理数 component  元素 determinant  行列式 rank  秩 overdetermined system  超定系统（即方程组无精确解） underdetermined system  不定系统（即方程组有无穷多解） orthonormal basis 正交基 null 零空间，核空间 l Some operations and functions ’ .’ \  rank det  inv rref null  l Application on solving linear systems of equations  Ax=b, A is an m×n matrix  ² Review:  ¸ b=0 Theory  r(A) =n,  there is only one exact solution zero.  r(A) <n,  there are infinite nonzero solutions.  Computation  r(A) =n,  only zero is its exact solution  r(A) <n,  row elementary transpositions→basis for the null space of A  ¸ b≠0 Theory  r(A)=r(A b)=n,  there is only one exact solution  r(A)=r(A b)<n,  there are infinite solutions  r(A)≠r(A b),  there is no exact solution  Computation  m=n & r(A)=r(A b)=n,  Cramer’ rule; A －1b; row elementary transposition  m≠n & r(A) =n,  row elementary transpositions  m≠n & r(A)=r(A b)<n  basis for the null space of A, particular solution