Linear Algebra

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Springer Science & Business Media, Sep 2, 1994 - Mathematics - 204 pages
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The original version of this book, handed out to my students in weekly in stallments, had a certain rugged charm. Now that it is dressed up as a Springer UTM volume, I feel very much like Alfred Dolittle at Eliza's wedding. I hope the reader will still sense the presence of a young lecturer, enthusiastically urging his audience to enjoy linear algebra. The book is structured in various ways. For example, you will find a test in each chapter; you may consider the material up to the test as basic and the material following the test as supplemental. In principle, it should be possible to go from the test directly to the basic material of the next chapter. Since I had a mixed audience of mathematics and physics students, I tried to give each group some special attention, which in the book results in certain sections being marked∑ "for physicists" or "for mathematicians. " Another structural feature of the text is its division into laconic main text, put in boxes, and more talkative unboxed side text. If you follow just the main text, jumping from box to box, you will find that it makes coherent reading, a real "book within the book," presenting all that I want to teach.
  

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Contents

Sets and Maps
x
12 Maps
4
13 Test
9
14 Remarks on the Literature
10
15 Exercises
11
Vector Spaces
13
22 Complex Numbers and Complex Vector Spaces
18
23 Vector Subspaces
22
66 Test
111
67 Determinant of an Endomorphism
113
68 The Leibniz Formula
114
69 Historical Aside
116
Systems of Linear Equations
118
72 Cramers Rule
120
73 Gaussian Elimination
122
74 Test
124

24 Test
23
25 Fields
25
26 What Are Vectors? A section for physicists
28
27 Complex Numbers 400 Years Ago Historical aside
37
28 Remarks on the Literature
38
Dimension
41
32 The Concept of Dimension
44
33 Test
48
34 Proof of the Basis Extension Theorem and the Exchange Lemma
49
35 The Vector Product A section for physicists
52
36 The Steinitz Exchange Theorem Historical aside
57
37 Exercises
58
Linear Maps
60
42 Matrices
66
43 Test
71
44 Quotient Spaces
73
45 Rotations and Reflections of the Plane A section for physicists
76
46 Historical Aside
80
Matrix Calculus
83
52 The Rank of a Matrix
87
53 Elementary Transformations
88
54 Test
91
55 How Does One Invert a Matrix? A section for mathematicians
92
56 Rotations and Reflections Continued
95
57 Historical Aside
98
58 Exercises
99
Determinants
101
62 Determination of Determinants
105
63 The Determinant of the Transposed Matrix
106
64 Determinantal Formula for the Inverse Matrix
108
65 Determinants and Matrix Products
110
75 More on Systems of Linear Equations
126
76 Captured on Camera A section for physicists
128
77 Historical Aside
131
79 Exercises
132
Euclidean Vector Spaces
134
82 Orthogonal Vectors
137
83 Orthogonal Maps
141
84 Groups
142
85 Test
144
86 Remark on the Literature
146
Eigenvalues
149
92 The Characteristic Polynomial
152
93 Test
154
94 Polynomials
156
95 Exercises
159
The Principal Axes Transformation
160
102 Symmetric Matrices
161
103 The Principal Axes Transformation for SelfAdjoint Endomorphisms
164
104 Test
166
105 Exercises Exercises for mathematicians
168
Classification of Matrices
169
112 The Rank Theorem
172
113 The Jordan Normal Form
174
114 More on the Principal Axes Transformation
176
116 Test
181
117 Exercises
183
Answers to the Tests
185
References
198
Index
199
Copyright

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