ALGEBRA: A TEXT-BOOK OF DETERMINANTS, MATRICES, AND ALGEBRAIC FORMS
CONTENTS LINEAR SYSTEMS INTRODUCTION TO LINEAR SYSTEMS LINEAR EQUATIONS DEFINITION OF LINEAR SYSTEM SOLUTION OF LINEAR SYSTEM GEOMETRY OF SOLUTIONS IN TWO VARIABLES BACK-SUBSTITUTION INTRODUCTION TO GAUSS ELIMINATION GEOMETRY OF SOLUTIONS IN THREE VARIABLES LINEAR SYSTEMS WITH COMPLEX NUMBERS INTERCHANGES IN TERMS ELIMINATIONS AND SCALINGS GAUSS ELIMINATION MATRICES IN ECHELON FORM THE GAUSS ELIMINATION ALGORITHM SOLUTION ALGORITHM FOR LINEAR SYSTEMS THE GAUSS–JORDAN ELIMINATION ALGORITHM EXISTENCE AND UNIQUENESS OF SOLUTIONS HOMOGENEOUS LINEAR SYSTEMS NUMERICAL CONSIDERATIONS APPLICATIONS: ECONOMICS, CHEMISTRY, PHYSICS, ENGINEERING ECONOMICS: THE DEMAND FUNCTION, MARKET EQUILIBRIA CHEMISTRY: CHEMICAL SOLUTIONS, BALANCING OF REACTIONS PHYSICS AND ENGINEERING: CIRCUITS, HEAT CONDUCTION TRAFFIC FLOW STATICS AND WEIGHT BALANCING NUMERICAL SOLUTIONS OF LINEAR SYSTEMS COMPUTATIONAL EFFICIENCY OF ROW REDUCTION ITERATIVE METHODS JACOBI ITERATION GAUSS–SEIDEL ITERATION CONVERGENCE COMPARISON OF ELIMINATION AND GAUSS–SEIDEL ITERATION NUMERICAL CONSIDERATIONS: ILL-CONDITIONING AND PIVOTING MINIPROJECTS PSYCHOLOGY: ANIMAL INTELLIGENCE COUNTING OPERATIONS IN GAUSS ELIMINATION ARCHIMEDES’ CATTLE PROBLEM TECHNOLOGY-AIDED PROBLEMS AND ANSWERS SELECTED SOLUTIONS WITH MATHEMATICA SELECTED SOLUTIONS WITH MATLAB SELECTED SOLUTIONS WITH MAPLE VECTORS MATRICES AND VECTORS ADDITION AND SCALAR MULTIPLICATION TRANSPOSE, SYMMETRIC AND HERMITIAN MATRICES SPECIAL SQUARE MATRICES, TRACE GEOMETRIC INTERPRETATION OF VECTORS APPLICATION OF LINEAR COMBINATIONS DIGITAL SIGNALS MATRIX TRANSFORMATIONS THE MATRIX–VECTOR PRODUCT MATRIX FORM OF LINEAR SYSTEMS RELATION BETWEEN THE SOLUTIONS OF AX = 0 AND AX = B THE SPAN LINEAR INDEPENDENCE DOT PRODUCT, LINES, HYPERPLANES DOT PRODUCT ORTHOGONAL PROJECTIONS LINES, PLANES, AND HYPERPLANES HYPERPLANES AND SOLUTIONS OF LINEAR SYSTEMS APPLICATION: COMPUTER GRAPHICS PLANE MATRIX TRANSFORMATIONS SPACE MATRIX TRANSFORMATIONS AFFINE TRANSFORMATIONS APPLICATIONS: AVERAGING, DYNAMICAL SYSTEMS DATA SMOOTHING BY AVERAGING DISCRETE DYNAMICAL SYSTEMS A POPULATION GROWTH MODEL SPECIAL TOPIC: TESSELLATIONS IN WEATHER MODELS MINIPROJECT: SPECIAL AFFINE TRANSFORMATIONS MATRICES MATRIX MULTIPLICATION ANOTHER VIEWPOINT OF MATRIX MULTIPLICATION POWERS OF A SQUARE MATRIX MATRIX MULTIPLICATION WITH COMPLEX NUMBERS MOTIVATION FOR MATRIX MULTIPLICATION COMPUTATIONAL CONSIDERATIONS APPLICATION TO COMPUTER GRAPHICS APPLICATION TO MANUFACTURING MATRIX INVERSE COMPUTATION OF THE INVERSE RELATION OF A?1 TO SQUARE SYSTEMS PROPERTIES OF MATRIX INVERSION MATRIX POWERS WITH NEGATIVE EXPONENTS APPLICATION TO STATICS: STIFFNESS OF ELASTIC BEAM ELEMENTARY MATRICES ELEMENTARY MATRICES AND INVERTIBLE MATRICES THE MATRIX INVERSION ALGORITHM CHARACTERIZATION OF INVERTIBLE MATRICES LU FACTORIZATION COMPUTATIONAL EFFICIENCY WITH LU LU WITH INTERCHANGES BLOCK AND SPARSE MATRICES BLOCK MATRICES ADDITION OF BLOCK MATRICES MULTIPLICATION OF BLOCK MATRICES INVERSION OF BLOCK MATRICES SPARSE MATRICES APPLICATIONS: LEONTIEF MODELS, MARKOV CHAINS STOCHASTIC MATRICES ECONOMICS: LEONTIEF INPUT–OUTPUT MODELS PROBABILITY MATRICES AND MARKOV PROCESSES GRAPH THEORY GRAPHS SOCIOLOGY AND PSYCHOLOGY: DOMINANCE GRAPHS CRYPTOLOGY: THE HILL CIPHER TRANSITION OF PROBABILITIES DIGRAPH WALKS A THEORETICAL PROBLEM VECTOR SPACES VECTOR SPACE DEFINITION AND PROPERTIES EXAMPLES OF VECTOR SPACES SUBSPACES COMPLEX VECTOR SPACES SPAN, LINEAR INDEPENDENCE SPAN LINEAR DEPENDENCE LINEAR DEPENDENCE FOR SEQUENCES BASIS, DIMENSION BASIS OF A VECTOR SPACE DIMENSION ORDERED BASES COORDINATES, CHANGE OF BASIS COORDINATE VECTORS CHANGE OF BASIS NULL SPACE COLUMN SPACE, ROW SPACE, RANK THE COLUMN SPACE THE ROW SPACE RANK RANK AND LINEAR SYSTEMS APPLICATION TO CODING THEORY VECTOR SPACES OVER Z2 THE HAMMING (7, 4)-CODE ENCODING AND DECODING OTHER TYPES OF CODES LINEAR TRANSFORMATIONS EVALUATION OF LINEAR TRANSFORMATION FROM A BASIS KERNEL AND RANGE ISOMORPHISMS MATRIX OF LINEAR TRANSFORMATION CHANGE OF BASIS AND THE MATRIX OF A LINEAR TRANSFORMATION PROOF OF DIMENSION THEOREM THE ALGEBRA OF LINEAR TRANSFORMATIONS SUMS AND SCALAR PRODUCTS COMPOSITION OF LINEAR TRANSFORMATIONS PROJECTIONS LINEAR TRANSFORMATION AND MATRIX OPERATIONS INVERTIBLE LINEAR TRANSFORMATIONS SPECIAL TOPIC: FRACTALS FRACTALS MINIPROJECT ANOTHER FRACTAL DETERMINANTS DETERMINANTS: BASIC CONCEPTS COFACTOR EXPANSION GEOMETRIC PROPERTY OF THE DETERMINANT THE SARRUS SCHEME FOR 3 × 3 DETERMINANTS PROPERTIES OF DETERMINANTS ELEMENTARY OPERATIONS AND DETERMINANTS MATRIX OPERATIONS AND DETERMINANTS THE ADJOINT; CRAMER’S RULE ADJOINT AND INVERSE CRAMER’S RULE DETERMINANTS WITH PERMUTATIONS PERMUTATIONS COMPUTATIONAL CONSIDERATION APPLICATIONS: GEOMETRY, POLYNOMIAL SYSTEMS EQUATIONS OF GEOMETRIC OBJECTS ELIMINATION THEORY, RESULTANTS, AND POLYNOMIAL SYSTEMS SPECIAL TOPIC: IMAGE RECOGNITION INTRODUCTION TO PROJECTIVE GEOMETRY PROJECTIVE TRANSFORMATIONS PROJECTIVE INVARIANTS THE OBJECT-IMAGE EQUATIONS VANDERMONDE DETERMINANTS BEZOUT RESULTANT EIGENVALUES EIGENVALUES AND EIGENVECTORS COMPUTATION OF EIGENVALUES AND EIGENVECTORS EIGENVALUES OF LINEAR OPERATORS NUMERICAL NOTE DIAGONALIZATION DIAGONALIZATION OF MATRICES POWERS OF DIAGONALIZABLE MATRICES AN IMPORTANT CHANGE OF VARIABLES APPLICATIONS: DISCRETE DYNAMICAL SYSTEMS BASIC CONCEPTS LONG-TERM BEHAVIOR OF DYNAMICAL SYSTEMS UNCOUPLING DYNAMICAL SYSTEMS APPLICATIONS: DYNAMICAL SYSTEMS (2) AND MARKOV CHAINS DYNAMICAL SYSTEMS WITH COMPLEX EIGENVALUES APPLICATION TO A POPULATION GROWTH PROBLEM MARKOV CHAINS AND STOCHASTIC MATRICES LIMITS OF STOCHASTIC MATRICES SPECIAL TOPIC: THE PAGERANK ALGORITHM OF GOOGLE APPROXIMATIONS OF EIGENVALUES THE POWER METHOD RAYLEIGH QUOTIENTS (THE RAYLEIGH–RITZ METHOD) ORIGIN SHIFTS INVERSE POWER METHOD SHIFTED INVERSE POWER METHOD APPLICATION TO ROOTS OF POLYNOMIALS THE CAYLEY–HAMILTON THEOREM GERSCHGORIN CIRCLES TRANSITION OF PROBABILITIES (PART 2) ORTHOGONALITY AND LEAST SQUARES ORTHOGONAL SETS AND MATRICES ORTHOGONAL SETS ORTHONORMAL SETS ORTHOGONAL MATRICES WHAT MAKES ORTHOGONAL MATRICES IMPORTANT THE GRAM–SCHMIDT PROCESS ORTHOGONAL COMPLEMENTS ORTHOGONAL PROJECTIONS AND BEST APPROXIMATION DISTANCE AND ANGLE FROM VECTOR TO SUBSPACE THE QR FACTORIZATION THE QR METHOD FOR EIGENVALUES HOUSEHOLDER TRANSFORMATIONS AND QR LEAST SQUARES A LEAST SQUARES PROBLEM SOLUTION OF THE LEAST SQUARES PROBLEM LEAST SQUARES WITH QR FACTORIZATION INNER PRODUCT SPACES DEFINITION OF INNER PRODUCT EXAMPLES OF INNER PRODUCTS LENGTH AND ORTHOGONALITY BASIC IDENTITIES AND INEQUALITIES COMPLEX INNER PRODUCTS; UNITARY MATRICES DEFINITION AND EXAMPLES UNITARY MATRICES POLYNOMIAL AND CONTINUOUS LEAST SQUARES POLYNOMIAL LEAST SQUARES CONTINUOUS LEAST SQUARES (REQUIRES CALCULUS) SPECIAL TOPIC: THE NFL RATING OF QUARTERBACKS THE PAULI SPIN AND DIRAC MATRICES RIGID MOTIONS IN RN VOLUME OF THE PARALLELEPIPED AND THE GRAM DETERMINANT QUADRATIC FORMS, SVD, WAVELETS ORTHOGONALIZATION OF SYMMETRIC MATRICES PROOF OF SCHUR’S DECOMPOSITION THEOREM AND EXAMPLE QUADRATIC FORMS AND CONIC SECTIONS DIAGONALIZATION OF QUADRATIC FORMS APPLICATIONS OF QUADRATIC FORMS TO GEOMETRY POSITIVE AND NEGATIVE DEFINITE QUADRATIC FORMS THE SINGULAR VALUE DECOMPOSITION (SVD) SINGULAR VALUES; FINDING V, ?, AND U PSEUDOINVERSE SVD AND LEAST SQUARES THE POLAR DECOMPOSITION OF A SQUARE MATRIX SPECIAL TOPIC: SVD AND IMAGE COMPRESSION FOURIER SERIES AND POLYNOMIALS APPLICATION TO WAVELETS WAVELETS AN IMAGE COMPRESSION PROJECT GENERATED BY CHATGPT INTERNATIONAL LICENSE WILLIAM GILBERT STRANG (BORN IN 1934) IS AN AMER- ICAN MATHEMATICIAN KNOWN FOR HIS CONTRIBUTIONS TO NUMERICAL LINEAR ALGEBRA, FINITE ELEMENT THE- ORY, THE CALCULUS OF VARIATIONS, AND WAVELETS HE IS A RENOWNED MATHEMATICS EDUCATOR WHO HAS AUTHORED SEVERAL INFLUENTIAL TEXTBOOKS ON LINEAR ALGEBRA INTRODUCTION LINEAR SYSTEMS, VECTORS, AND MATRICES ARE THE THREE PILLARS OF LINEAR ALGEBRA IN THIS CHAPTER, WE DISCUSS LINEAR SYSTEMS THESE ARE SETS OF LINEAR EQUATIONS A LIN- EAR EQUATION IS THE SIMPLEST FORM OF A MATHEMATICAL EQUATION WE CAN IMAGINE THE UN- KNOWNS ARE MULTIPLIED BY GIVEN NUMBERS AND ADDED TOGETHER TO YIELD ANOTHER GIVEN NUMBER THE UNKNOWNS ARE IN THE FIRST POWER; THERE ARE NO PRODUCTS OR POWERS OR FRAC- TIONS OR ANY OTHER FUNCTIONS IN THEM LINEAR SYSTEMS HAVE BEEN STUDIED FOR CENTURIES THE THIRD CENTURY BCE CHINESE TEXT NINE CHAPTERS OF MATHEMATICAL ART DEVOTES ITS EIGHTH CHAPTER (TITLED FANGCHENG) TO EIGH- TEEN WORD PROBLEMS THAT RESULT IN LINEAR SYSTEMS, WHICH ARE THEN SOLVED (FIGURE 1 2) THE FIRST PROBLEM IS THE FOLLOWING [15, 16] SUPPOSE THAT H STANDS FOR “THE NUMBER OF UNITS OF HIGH-QUALITY RICE STRAWS”, LIKEWISE, M FOR “MID-QUALITY”, AND L FOR “LOW-QUALITY” FIND H, M, AND L GIVEN THAT: – 3 BUNDLES OF H, 2 BUNDLES OF M, AND 1 BUNDLE OF L PRODUCE 39 UNITS OF RICE – 2 BUNDLES OF H, 3 BUNDLES OF M, AND 1 BUNDLE OF L PRODUCE 34 UNITS OF RICE – 1 BUNDLE OF H, 2 BUNDLES OF M, AND 3 BUNDLES OF L PRODUCE 26 UNITS OF RICE THE ABOVE INFORMATION LEADS TO THE FOLLOWING SYSTEM OF LINEAR EQUATIONS: 3H + 2M + L = 39, 2H + 3M + L = 34, H + 2M + 3L = 26, WHICH NEEDS TO BE SOLVED FOR H, M, AND L 1 LINEAR SYSTEMS FIGURE 1 2: A PAGE OF THE NINE CHAPTERS ON MATHEMATICAL ART (1820 EDITION) (HTTP://PMGS KONGFZ COM/DETAIL/1_158470/, PUBLIC DOMAIN, HTTPS://COMMONS WIKIMEDIA ORG/W/INDEX PHP?CURID=22913440 ) THE “NINE CHAPTERS ON THE MATHEMATICAL ART” IS AN INFLUENTIAL ANCIENT CHINESE MATHEMATICAL TREATISE, WHICH IS A COMPILATION OF MATHEMATI- CAL PROBLEMS AND PROBLEM SOLVING TECHNIQUES THAT WERE PRACTICED AND REFINED OVER SEVERAL CENTURIES IT IS BELIEVED TO HAVE BEEN WRITTEN AND EDITED OVER A LONG PERIOD, FROM AROUND THE TENTH CENTURY BCE TO THE CE SECOND CENTURY, DURING CHINA’S HAN DYNASTY ANOTHER INTERESTING OLD PROBLEM RESULTING IN SOLVING A LINEAR SYSTEM IS THE ARCHIME- DES CATTLE PROBLEM THIS IS DISCUSSED IN SECTION 1 5 MODERN USES OF LINEAR SYSTEMS ARE DISCUSSED IN SECTION 1 3 WITH APPLICATIONS TO ECO- NOMICS (THE DEMAND FUNCTION), CHEMISTRY (BALANCING OF CHEMICAL REACTIONS), PHYSICS AND ENGINEERING (ELECTRICAL NETWORKS, HEAT CONDUCTION, STATICS AND WEIGHT BALANCING), AND TRAFFIC FLOW PROBLEMS 1 1 INTRODUCTION TO LINEAR SYSTEMS WE INTRODUCE LINEAR SYSTEMS AND STUDY A METHOD OF SOLUTION CALLED ELIMINATION, OR GAUSS ELIMINATION THIS SECTION INTRODUCES THE METHOD, AND SECTION 1 2 OFFERS A MORE IN-DEPTH STUDY OF IT 1 1 1 LINEAR EQUATIONS THE FIRST EQUATIONS ENCOUNTERED IN SCHOOL ALGEBRA ARE LINEAR EQUATIONS A LINEAR EQUATION IN UNKNOWNS OR VARIABLES X1, , XN IS AN EQUATION THAT CAN BE WRITTEN IN THE STANDARD FORM A1X1 + A2X2 + ???+ ANXN = B (1 1) THE AIS ARE THE COEFFICIENTS, AND B IS THE CONSTANT TERM THESE ARE REAL NUMBERS AND, ON OCCASION, COMPLEX NUMBERS THE FIRST VARIABLE WITH NONZERO COEFFICIENT OF A LINEAR EQUATION IS CALLED THE LEADING VARIABLE THE REMAINING VARIABLES ARE CALLED FREE VARIABLES AN EQUATION THAT IS NOT LINEAR IS CALLED NONLINEAR EXAMPLE 1 1 1 1 THE EQUATION X1 + X2 + 4X3 ?6X4 ?1 = X1 ?X2 + 1 1 INTRODUCTION TO LINEAR SYSTEMS IS LINEAR, BECAUSE IT CAN BE WRITTEN IN THE FORM (1 1) AS 0X1 + 2X2 + 4X3 ?6X4 = 3 THE LEADING VARIABLE IS X2 THE FREE VARIABLES ARE X1, X3, AND X4 2 THE FOLLOWING EQUATIONS ARE LINEAR: X1 + 2X2 ??5X3 ?X4 = 0, X ?4Y + 9Z = TAN(4), F = + 6X3 = (1 3) + 6X3 = 0 A RECTANGULAR ARRANGEMENT OF ELEMENTS FROM A SET IS CALLED A MATRIX OVER THAT SET THESE ELEMENTS ARE THE ENTRIES OF THE MATRIX A MATRIX HAS ROWS, NUMBERED TOP TO BOT- TOM, AND COLUMNS, NUMBERED LEFT TO RIGHT THE ENTRY AT THE INTERSECTION OF THE ITH ROW AND JTH COLUMN IS THE (I, J) ENTRY A MATRIX WITH M ROWS AND N COLUMNS HAS SIZE M × N (PRONOUNCED “M BY N”) A MATRIX WITH ONLY ONE COLUMN IS ALSO CALLED A VECTOR USUALLY, THE MATRICES IN THIS TEXT HAVE ENTRIES THAT ARE REAL NUMBERS AND, SOMETIMES, COMPLEX NUMBERS THE MATRIX WHOSE ROWS CONSIST OF THE COEFFICIENTS AND CONSTANT TERMS OF EACH EQUA- TION OF A LINEAR SYSTEM IS CALLED THE AUGMENTED MATRIX OF THE SYSTEM THE AUGMENTED MATRIX OF SYSTEM (1 3) IS [[ [ ]] ] OR [[ [ ]] ] THE SECOND NOTATION INDICATES THAT THE LAST COLUMN CONSISTS OF CONSTANT TERMS THE MATRIX WITH ENTRIES THE COEFFICIENTS OF THE SYSTEM IS THE COEFFICIENT MATRIX THE VECTOR OF ALL CON- STANT TERMS IS THE VECTOR OF CONSTANTS THE COEFFICIENT MATRIX AND THE VECTOR OF CONSTANTS OF SYSTEM (1 3) ARE [[ [