APPLIED LINEAR ALGEBRA

By H. FADAR

ASSIGNMENTS MATRICES LISTS OF LISTS MAPS 1-1 AND ONTO SOME EXAMPLES MAPS AND THEIR GRAPHS INVERTIBILITY THE INVERSION OF MAPS 2 VECTOR SPACES AND LINEAR MAPS VECTOR SPACES, ESPECIALLY SPACES OF FUNCTIONS LINEAR MAPS LINEAR MAPS FROM IFN THE LINEAR EQUATION A? = Y, AND RAN A AND NULL A INVERSES 3 ELIMINATION, OR: THE DETERMINATION OF NULL A AND RAN A ELIMINATION AND BACKSUBSTITUTION THE REALLY REDUCED ROW ECHELON FORM AND OTHER REDUCED FORMS A COMPLETE DESCRIPTION FOR NULL A OBTAINED FROM A B-FORM THE FACTORIZATION A = A(:, BOUND)RRREF(A) A ‘BASIS’ FOR RAN A UNIQUENESS OF THE RRREF(A) THE RRREF(A) AND THE SOLVING OF A? = Y THE PIGEONHOLE PRINCIPLE FOR SQUARE MATRICES 4 THE DIMENSION OF A VECTOR SPACE BASES CONSTRUCTION OF A BASIS DIMENSION SOME USES OF THE DIMENSION CONCEPT THE DIMENSION OF IFT DIRECT SUMS ELIMINATION IN VECTOR SPACES 5 THE INVERSE OF A BASIS, AND INTERPOLATION DATA MAPS A FORMULA FOR THE COORDINATE MAP CHANGE OF BASIS INTERPOLATION AND LINEAR PROJECTORS 6 INNER PRODUCT SPACES DE?NITION AND EXAMPLES THE CONJUGATE TRANSPOSE ORTHOGONAL PROJECTORS AND CLOSEST POINTS LEAST-SQUARES ORTHONORMAL COLUMN MAPS RAN A AND NULL AC FORM AN ORTHOGONAL DIRECT SUM FOR TAR A THE INNER PRODUCT SPACE IFM×N AND THE TRACE OF A MATRIX 7 NORMS, MAP NORMS, AND THE CONDITION OF A BASIS HOW TO JUDGE THE ERROR BY THE RESIDUAL THE MAP NORM VECTOR NORMS AND THEIR ASSOCIATED MAP NORMS 8 FACTORIZATION AND RANK THE NEED FOR FACTORING LINEAR MAPS THE TRACE OF A LINEAR MAP THE RANK OF A MATRIX AND OF ITS (CONJUGATE) TRANSPOSE ELIMINATION AS FACTORIZATION SVD THE PSEUDO-INVERSE 2-NORM AND 2-CONDITION OF A MATRIX THE E?ECTIVE RANK OF A NOISY MATRIX THE POLAR DECOMPOSITION EQUIVALENCE AND SIMILARITY 9 DUALITY COMPLEMENTARY MATHEMATICAL CONCEPTS THE DUAL OF A VECTOR SPACE THE DUAL OF AN INNER PRODUCT SPACE THE DUAL OF A LINEAR MAP 10 THE POWERS OF A LINEAR MAP AND ITS SPECTRUM EXAMPLES EIGENVALUES AND EIGENVECTORS DIAGONA(LIZA)BILITY ARE ALL SQUARE MATRICES DIAGONABLE? DOES EVERY SQUARE MATRIX HAVE AN EIGENVALUE? POLYNOMIALS IN A LINEAR MAP IT IS ENOUGH TO UNDERSTAND THE EIGENSTRUCTURE OF MATRICES EVERY COMPLEX (SQUARE) MATRIX IS SIMILAR TO AN UPPER TRIANGULAR MATRIX 11 CONVERGENCE OF THE POWER SEQUENCE CONVERGENCE OF SEQUENCES IN A NORMED VECTOR SPACE THREE INTERESTING PROPERTIES OF THE POWER SEQUENCE OF A LINEAR MAP SPLITTING O?THE NONDEFECTIVE EIGENVALUES THREE INTERESTING PROPERTIES OF THE POWER SEQUENCE OF A LINEAR MAP: THE SEQUEL THE POWER METHOD 12 CANONICAL FORMS THE SCHUR FORM THE PRIMARY DECOMPOSITION THE JORDAN FORM 13 LOCALIZATION OF EIGENVALUES GERSHGORIN’S CIRCLES DETERMINANTS ANNIHILATING POLYNOMIALS THE MULTIPLICITIES OF AN EIGENVALUE PERRON-FROBENIUS 14 SOME APPLICATIONS 3-SPACE ROTATION IN 3-SPACE MARKOV CHAINS AN EXAMPLE FROM CAGD TRIDIAGONAL TOEPLITZ MATRIX LINEAR PROGRAMMING APPROXIMATION BY BROKEN LINES FLATS: POINTS, VECTORS, BARYCENTRIC COORDINATES, DI?ERENTIATION GRAD, DIV, AND CURL 15 OPTIMIZATION AND QUADRATIC FORMS MINIMIZATION QUADRATIC FORMS REDUCTION OF A QUADRATIC FORM TO A SUM OF SQUARES RAYLEIGH QUOTIENT 16 MORE ON DETERMINANTS DE?NITION AND BASIC PROPERTIES SYLVESTER CAUCHY-BINET