Thomas S. Shores Applied Linear Algebra and Matrix Analysis Contents 1 LINEAR SYSTEMS OF EQUATIONS .................... 1 SomeExamples......................................... 1 Notation and a Review of Numbers ........................ 9 GaussianElimination:BasicIdeas......................... 21 GaussianElimination:GeneralProcedure................... 33 *Computational Notes and Projects ....................... 46 2 MATRIX ALGEBRA...................................... 55 MatrixAdditionandScalarMultiplication.................. 55 MatrixMultiplication.................................... 62 ApplicationsofMatrixArithmetic......................... 71 SpecialMatricesandTransposes .......................... 86 MatrixInverses .........................................101 BasicPropertiesofDeterminants..........................114 *Computational Notes and Projects .......................129 3 VECTOR SPACES ........................................145 oncepts............................145 Subspaces ..............................................161 binations.....................................170 Subspaces Associated with Matrices and Operators ..........183 BasesandDimension ....................................191 LinearSystemsRevisited.................................199 *Computational Notes and Projects .......................208 4 GEOMETRICAL ASPECTS OF STANDARD SPACES ...211 StandardNormandInnerProduct ........................211 ApplicationsofNormsandInnerProducts..................221 Orthogonal and Unitary Matrices ..........................233 *Change of Basis and Linear Operators ....................242 *Computational Notes and Projects .......................247 xii Contents 5 THE EIGENVALUE PROBLEM ..........................251 DefinitionsandBasicProperties...........................251 Similarity and Diag
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