MATLAB Lecture 3 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr MATLAB Lecture 3--Linear Space 线性空间 Ref: matlab-Mathematics-Matrices and Linear algebra Solving linear Systems of equations ● Vocabulary Vector/linear space向量线性空间 linear relation线性关系 linear combination线性组合 linear expression/ representation线性表示 inearly dependence/correlation线性相关 linearly dependent线性相关的 linearly independence线性无关 linearly independent线性无关的 Linear space线性空间 linear spanning线性生成 dimension维数 linear subspace线性子空间 maximal linearly independent subset极大线性无关组 scalar数,标量 span生成,张成 basis ■■■■■■■a■■■■m■ factorization分解 metric matriⅸx对称矩阵 product乘积 triangular matⅸx三角矩阵 transpose转置 upper triangular matrix上三角阵 lower triangular matrix下三角阵 diagonal matrix对角阵 permutation置换 orthogonal matrix正交阵 unitary matrⅸx酉阵 operations and functions e Application on linear space ☆ Review V Let x,, x2,,x, be vectors in vector space V. A sum of the form k, x,+k2-x2+.+k,x where k,k,,.k, are scalars, is called a linear combination of x,, x,,, ,x, The set of all linear combinations of x, x2. is called the span of x, x,,],.The span of x,x2,…, x, will be denoted by Span(x1x2,…,xn) Y The vectors x,,x,,,x, in a vector space V are said to be linearly independent if kx+k2x2+…+knx implies that all the scalars k,, k,, ,k, must equal 0MATLAB Lecture 3  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec3­1 MATLAB Lecture 3 ­­ Linear Space 线性空间 Ref: MATLAB→Mathematics→Matrices and Linear Algebra  →Solving Linear Systems of Equations l Vocabulary: Vector/linear space 向量/线性空间 linear relation  线性关系 linear combination  线性组合 linear expression/representation  线性表示 linearly dependence/correlation  线性相关 linearly dependent  线性相关的 linearly independence 线性无关 linearly independent  线性无关的 Linear space 线性空间 linear spanning  线性生成 dimension  维数 linear subspace 线性子空间 maximal linearly independent subset  极大线性无关组 scalar 数，标量 span  生成，张成 basis  基 factorization  分解 symmetric matrix  对称矩阵 product  乘积 triangular matrix  三角矩阵 transpose 转置 upper triangular matrix  上三角阵 lower triangular matrix  下三角阵 diagonal matrix  对角阵 permutation  置换 orthogonal matrix  正交阵 unitary matrix  酉阵 l Some operations and functions rank rref rrefmovie null  l Application on linear space ² Review:  ¸ Let  1 2  , ,..., n  x x x be vectors in vector space V. A sum of the form  1 1 2 2  ... n n  k x + k x + + k x ,  where 1 2  , ,..., n  k k k are scalars, is called a linear combination of 1 2  , ,..., n  x x x . The set of all linear combinations of 1 2  , ,..., n  x x x is called the span of 1 2  , ,..., n  x x x . The span  of 1 2  , ,..., n  x x x will be denoted by Span 1 2  ( , ,..., ) n  x x x .  ¸ The vectors  1 2  , ,..., n  x x x in a vector space V are said to be linearly independent if  1 1 2 2  ... n n  k x + k x + + k x implies that all the scalars  1 2  , ,..., n  k k k must equal 0