Finite Dimensional Vector Spaces

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Princeton University Press, 1948 - Mathematics - 196 pages
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As a newly minted Ph.D., Paul Halmos came to the Institute for Advanced Study in 1938--even though he did not have a fellowship--to study among the many giants of mathematics who had recently joined the faculty. He eventually became John von Neumann's research assistant, and it was one of von Neumann's inspiring lectures that spurred Halmos to write Finite Dimensional Vector Spaces. The book brought him instant fame as an expositor of mathematics.

Finite Dimensional Vector Spaces combines algebra and geometry to discuss the three-dimensional area where vectors can be plotted. The book broke ground as the first formal introduction to linear algebra, a branch of modern mathematics that studies vectors and vector spaces. The book continues to exert its influence sixty years after publication, as linear algebra is now widely used, not only in mathematics but also in the natural and social sciences, for studying such subjects as weather problems, traffic flow, electronic circuits, and population genetics.

In 1983 Halmos received the coveted Steele Prize for exposition from the American Mathematical Society for "his many graduate texts in mathematics dealing with finite dimensional vector spaces, measure theory, ergodic theory, and Hilbert space."

 

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Contents

SPACES 1 Definition of vector space
1
2 Examples of vector spaces
2
3 Comments on notation and terminology
4
U Definition of linear dependence
5
5 Characterization of linear dependence
7
6 Definition and construction of bases
8
7 Dimension of a vector space
10
8 Isomorphism of vector spaces
12
17 Direct sums
27
18 Dimension of a direct sum
29
19 Conjugate spaces of direct sums
31
TRANSFORMATIONS 20 Definitions and examples of linear trans formations
33
21 Linear transformations as a vector space
35
22 Products of linear transformations
36
23 Polynomials in a linear transformation
37
2U Inverse of a linear transformation
39

9 Linear manifolds
14
10 Calculus of linear manifolds
15
11 Dimension of a linear manifold
17
12 Conjugate spaces
18
13 Notation for linear functionals
20
1U Bases in conjugate spaces
21
15 Reflexivity of finite dimensional spaces
23
16 Annihilators of linear manifolds
25
25 Definition of matrices
43
26 Isomorphism between matrices and operators
47
ORTHOGONALITY
86
THE CLASSICAL CANONICAL FORM
159
DIRECT PRODUCTS
170
HILBERT SPACE
183
Bibliography
189
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Nonlinear Programming
Olvi L. Mangasarian
No preview available - 1969
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