Contents Preface i x 1 Vecto r Spaces 1 1.1 Group s and fields 1 1.2 Vecto r spaces 4 1.3 Linea r dependence, bases, and dimension 1 4 1.4 System s of linear equations 2 2 *1.5 Norme d finite-dimensional linear spaces 3 2 2 Linea r Operators and Matrices 3 5 2.1 Linea r operators 3 5 2.2 Operato r multiplication 3 9 2.3 Matri x multiplication 4 1 2.4 Matrice s and operators 4 6 2.5 Kernel , range, nullity, and rank 5 1 *2.6 Operato r norms 5 6 3 Dualit y of Vector Spaces 5 7 3.1 Linea r functionals 5 7 3.2 Th e adjoint 6 2 4 Determinant s 6 5 4.1 Permutation s 6 5 4.2 Multilinea r maps 6 9 4.3 Alternatin g ra-forms 7 4 4.4 Determinan t of an operator 7 6 4.5 Determinan t of a matrix 7 9 V

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