1 LINEAR SYSTEMS OF EQUATIONS .................... 1
1.1 SomeExamples ......................................... 1
1.2 Notation and a Review of Numbers ........................ 9
1.3 GaussianElimination:BasicIdeas ......................... 21
1.4 GaussianElimination:GeneralProcedure................... 33
1.5 *Computational Notes and Projects ....................... 46
2 MATRIX ALGEBRA...................................... 55
2.1 MatrixAdditionandScalarMultiplication.................. 55
2.2 MatrixMultiplication.................................... 62
2.3 ApplicationsofMatrixArithmetic......................... 71
2.4 SpecialMatricesandTransposes .......................... 86
2.5 MatrixInverses .........................................101
2.6 BasicPropertiesofDeterminants ..........................114
2.7 *Computational Notes and Projects .......................129
3 VECTOR SPACES ........................................145
3.1 De?nitionsandBasicConcepts............................145
3.2 Subspaces ..............................................161
3.3 LinearCombinations.....................................170
3.4 Subspaces Associated with Matrices and Operators ..........183
3.5 BasesandDimension ....................................191
3.6 LinearSystemsRevisited.................................199
3.7 *Computational Notes and Projects .......................208
4 GEOMETRICAL ASPECTS OF STANDARD SPACES ...211
4.1 StandardNormandInnerProduct ........................211
4.2 ApplicationsofNormsandInnerProducts..................221
4.3 Orthogonal and Unitary Matrices ..........................233
4.4 *Change of Basis and Linear Operators ....................242
4.5 *Computational Notes and Projects .......................247
5 THE EIGENVALUE PROBLEM ..........................251
5.1 De?nitionsandBasicProperties...........................251
5.2 Similarity and Diagonalization ............................263
5.3 Applications toDiscreteDynamicalSystems ................272
5.4 Orthogonal Diagonalization ...............................282
5.5 *SchurFormandApplications ............................287
5.6 *TheSingularValueDecomposition........................291
5.7 *Computational Notes and Projects .......................294
6 GEOMETRICAL ASPECTS OF ABSTRACT SPACES ...305
6.1 NormedSpaces..........................................305
6.2 InnerProductSpaces ....................................312
6.3 Gram–SchmidtAlgorithm ................................323
6.4 LinearSystemsRevisited.................................333
6.5 *Operator Norms ........................................342
6.6 *Computational Notes and Projects .......................348
Table of Symbols ..............................................355
Solutions to Selected Exercises ................................357
References .....................................................375
Index ..........................................................377