Linear Algebra: A Pure Mathematical Approach

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Springer 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.
 

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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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