Linear Algebra: A Pure Mathematical ApproachSpringer Science & Business Media, 2002 M10 1 - 250 páginas In algebra, an entity is called linear if it can be expressed in terms of addition, and multiplication by a scalar; a linear expression is a sum of scalar multiples of the entities under consideration. Also, an operation is called linear if it preserves addition, and multiplication by a scalar. For example, if A and Bare 2 x 2 real matrices, v is a (row) vector in the real plane, and c is a real number, then v(A + B) = vA + vB and (cv)A = c(vA), that is, the process of applying a matrix to a vector is linear. Linear Algebra is the study of properties and systems which preserve these two operations, and the following pages present the basic theory and results of this important branch of pure mathematics. There are many books on linear algebra in the bookshops and libraries of the world, so why write another? A number of excellent texts were written about fifty years ago (see the bibliography); in the intervening period the 'style' of math ematical presentation has changed. Also, some of the more modern texts have concentrated on applications both inside and outside mathematics. There is noth ing wrong with this approach; these books serve a very useful purpose. But linear algebra contains some fine pure mathematics and so a modern text taking the pure mathematician's viewpoint was thought to be worthwhile. |
Contenido
Algebraic Preamble | 1 |
Permutation Groups | 6 |
Problems 1 | 9 |
Vector Spaces and Linear Maps | 13 |
Bases and Dimension | 17 |
Linear Maps | 20 |
Direct Sums | 24 |
Addendum Modules | 27 |
Problems 5 | 115 |
Hermitian and Inner Product Spaces | 117 |
Hermitian and Inner Products and Norms | 118 |
Unitary and Selfadjoint Maps | 131 |
Orthogonal and Symmetric Maps | 139 |
Problems 6 | 143 |
Selected Topics | 149 |
Normed Algebras Quaternions and Cayley Numbers | 159 |
Problems 2 | 27 |
Matrices Determinants and Linear Equations | 33 |
Determinants | 45 |
Systems of Linear Equations | 56 |
Problems 3 | 59 |
CayleyHamilton Theorem and Jordan Form | 67 |
The CayleyHamilton and Spectral Theorems | 73 |
Jordan Form | 84 |
Problems 4 | 95 |
Interlude on Finite Fields | 101 |
Applications | 108 |
Introduction to the Representation of Finite Groups | 168 |
Problems 7 | 180 |
Set Theory | 185 |
Problems A | 192 |
Answers and Solutions to the Problems | 195 |
Bibliography | 221 |
Notation index | 223 |
Definition Index | 225 |
Theorem Index | 226 |
Subject Index | 227 |
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Términos y frases comunes
2-cycles a₁ associated axioms B₁ bijection c₁ called Cayley-Hamilton Theorem Chapter characteristic polynomial coefficient column commutative Corollary defined denoted determinant diagonal entries dimension direct sum E₁ eigenvalues eigenvectors elements End(V equals equivalence example factor finite field finite-dimensional function G-module G-submodule given in Problem gives Hence Hermitian product homomorphism identity map Inequality injective inner product space inverse irreducible isomorphism Jordan Form Lemma linear algebra linear combination linear equations linear map linearly independent matrix multiplication minimum polynomial modulo n x n non-singular non-zero normed algebra Note orthogonal orthonormal basis permutation positive definite positive integer properties prove quadratic form reader should check real numbers result follows ring satisfying scalar multiplication Secondly self-adjoint similar solution spans spectral standard basis subgroup subset subspace superdiagonal Suppose surjective symmetric theory u₁ unique unitary V₁ vector space w₁ zero