MATLAB Lecture 3 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr >>A=fix(10*rand(3, 2)): %Obtain a set of vectors which are the columns ofA >>X=fx(10°rand(3,1)) >>r0=rank(A ); >>r l=rank (AxD) >>ifro=rl disp can be written by the columns ofA else disp 'x cannot be written by the columns of A d basis of a vector space >>A=fix(10*rand( 3, 4)): %Obtain a set of vectors which are the columns of A >>B=rref(a) Observe the columns of b we may obtain the basis of the linear spanning space of columns ofA > rrefmovie(A) Movie of the computation of the reduced row echelon form .*Advanced study for linear system 令 Cholesky,LU, and QR Factorizations(分解) tic toc All three of these factorizations make use of triangular matrices where all the elements either above or below the diagonal are zero. Systems of linear equations involving triangular matrices are easily and quickly sol ved using either forward or back substitution. The Cholesky factorization expresses a symmetric matrix(对称矩阵) as the product of a triangular matrix and its transpose A=RR, where r is an upper triangular matrix LU factorization, or Gaussian elimination, expresses any square matrix A as the product of a permutation of a lower triangular matrix and an upper triangular matrix A=LU where L is a permutation of a lower triangular matrix with ones on its diagonal and U is an upper triangular matrix The orthogonal, or QR, factorization expresses any rectangular matrix as the product of an orthogonal or unitary matrix and an upper triangular matrix. A column permutation may also be involved. A=ORor AP=OR where g is orthogonal or unitary, R is upper triangular, and P is a permutation. An orthogonal matrix or a matrix with orthonormal columns. is a real matrix whose columns all have unit length and are perpendicular to each other. If g is orthogonal, then 00=7 For complex matrices, the corresponding term is unitaryMATLAB Lecture 3  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec3­3  >> A = fix (10*rand ( 3, 2 ) );  % Obtain a set of vectors which are the columns of A  >> x = fix (10*rand ( 3,1 ) );  >> r 0 = rank ( A );  >> r 1= rank ([A x]);  >> if r 0 == r1  disp ‘x can be written by the columns of A.’ else disp ‘x cannot be written by the columns of A.’ end  basis of a vector space >> A = fix (10*rand ( 3, 4 ) );  % Obtain a set of vectors which are the columns of A  >> B = rref (A) % Observe the columns of B we may obtain the basis of the linear …  spanning space of columns of A  >> rrefmovie (A) % Movie of the computation of the reduced row echelon form  l *Advanced study for linear system ² Cholesky, LU, and QR Factorizations  (分解) lu qr chol  tic toc ² Theorem  All three of these factorizations make use of triangular matrices where all the elements either above or below the diagonal  are zero.  Systems  of linear equations  involving triangular matrices are easily and quickly solved using either forward or back substitution.  The Cholesky factorization expresses a symmetric matrix（对称矩阵）as the product of a triangular matrix and its transpose A = R¢ R , where R is an upper triangular matrix. LU factorization, or Gaussian elimination, expresses any square matrix A as the product of a permutation of a lower triangular matrix and an upper triangular matrix A = LU where L is a permutation of a lower triangular matrix  with ones  on its  diagonal  and U is an upper triangular matrix.  The orthogonal, or QR, factorization expresses any rectangular matrix as the product of an  orthogonal or unitary matrix and an upper triangular matrix. A column permutation may also  be involved. A = QR or AP = QR where Q is orthogonal or unitary, R is upper triangular,  and P is a permutation.  An orthogonal matrix, or a matrix with orthonormal columns, is a real matrix whose columns  all have unit length and are perpendicular to each other. If Q is orthogonal, then Q¢Q = I .  For complex matrices, the corresponding term is unitary