Basic Linear Algebra

1 MATRICES 1 1.1 SCALARS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 OPERATIONS ON MATRICES . . . . . . . . . . . . . . . . . 10 1.3.1 MULTIPLICATION OF PARTITIONED MATRICES . . . . . . . . . . . . . . 15 1.4 SPECIAL TYPES OF MATRICES . . . . . . . . . . . . . . . 24 1.5 ROW EQUIVALENCE, INVERTIBILITY . . . . . . . . . . . 32 1.5.1 ELEMENTARY ROW OPERATIONS, ROW EQUIVALENCE . . . . . . . . . . . . . . . . . . 32 2 SYSTEMS OF LINEAR EQUATIONS 51 2.1 SYSTEMS OF LINEAR EQUATIONS . . . . . . . . . . . . . 51 2.2 SYSTEMS OF HOMOGENEOUS EQUATIONS . . . . . 57 2.2.1 INVERTIBILITY AND SYSTEMS OF LINEAR EQUATIONS . . . . . . . . . . . . . . . . . . 61 3 DETERMINANTS 73 3.1 DEFINITION OF DETERMINANTS . . . . . . . . . . . . . . 73 3.2 COFACTOR EXPANSIONS AND COMPUTATIONAL PROPERTIES . . . . . . . . . . . . . . 84 3.2.1 An Application: CRAMERíS RULE . . . . . . . . . 87 3.2.2 TRACE . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4 VECTOR SPACES 107 4.1 VECTOR SPACES . . . . . . . . . . . . . . . . . . . . . . . . 107 v vi CONTENTS 4.1.1 SUBSPACES . . . . . . . . . . . . . . . . . . . . . . . 112 4.1.2 LINEAR SPAN . . . . . . . . . . . . . . . . . . . . . . 115 4.2 LINEAR DEPENDENCE, INDEPENDENCE, BASES . . . . . . . . . . . . . . . . . . . 124 4.3 BASIS AND DIMENSION . . . . . . . . . . . . . . . . . . . . 127 4.3.1 STANDARD BASES FOR SOME VECTOR SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.3.2 CONSTRUCTION OF BASES, DIMENSION . . . . . 130 4.4 COORDINATES . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.5 SUBSPACES ASSOCIATED WITH A MATRIX . . . . . . . . . . . . . . . . . . . . . . . 144 4.5.1 ROW SPACE . . . . . . . . . . . . . . . . . . . . . . . 144 4.5.2 COLUMN SPACE . . . . . . . . . . . . . . . . . . . . 147 4.5.3 THE SOLUTION SPACE OF AX = O . . . . . . . . . 151 5 INNER PRODUCT SPACES 161 5.1 INNER PRODUCTS . . . . . . . . . . . . . . . . . . . . . . . 161 5.1.1 NATURAL EXAMPLES OF REAL INNER PRODUCT SPACES . . . . . . . . . . . . . . . . . . . 165 5.2 NORM AND ORTHOGONALITY . . . . . . . . . . . . . . . 171 5.3 ORTHOGONAL AND ORTHONORMAL BASES . . . . . . . 180 5.3.1 THE GRAM-SCHMIDT ORTHOGONALIZATION PROCESS . . . . . . . . . . 183 5.3.2 ORTHOGONAL PROJECTIONS . . . . . . . . . . . . 184 5.3.3 Application 1: The Method of Least Squares . . . . 189 5.3.4 Application 2: FOURIER SERIES . . . . . . . . . . 193 6 DIAGONALIZATION AND ITS APPLICATIONS 203 6.1 EIGENVALUES, EIGENVECTORS AND DIAGONALIZATION . . . . . . . . . . . . . . . . . . . . . . 203 6.1.1 MATRIX EXPONENTIALS . . . . . . . . . . . . . . . 208 6.1.2 Application: SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . 215 6.1.3 CAYLEY-HAMILTON THEOREM . . . . . . . . . . . 225 6.2 DIAGONALIZATION OF REAL SYMMETRIC MATRICES . . . . . . . . . . . . . . . . . . . 232 6.2.1 Application: QUADRATIC FORMS . . . . . . . . . 237 CONTENTS vii 7 LINEAR TRANSFORMATIONS 247 7.1 DEFINITION OF LINEAR TRANSFORMATIONS 7.2 MATRIX REPRESENTATION OF LINEAR TRANSFORMATIONS