The present volume is the second in the author's series of three dealing with abstract algebra. For an understanding of this volume a certain familiarity with the basic concepts treated in Volume I: groups， rings， fields， homomorphisms， is presup- posed. However， we have tried to make this account' of linear algebra independent of a detailed knowledge of our first volume. References to specific results are given occasionally but some of the fundamental concepts needed have been treated again. In short， it is hoped that this volume can be read with complete understanding by any student who is mathematically sufficiently mature and who has a familiarity with the standard notions of modern algebra.
CHAPTER I: FINITE DIMENSIONAL VECTOR SPACES
S ECTION
1. Abstract vector spaces
2. Rightvectorspaces
3. o-modules
4. Linear dependence
5. Invariance of dimensionality
6. Bases and matrices
7. Applications to matrix theory
8. Rank ofa set ofvectors
9. Factor spaces
10. Algebra ofsubspaces
11. Independent subspaces, direct sums
CHAPTER II: LINEAR TRANSFORMATIONS
1. Definition and examples
2. Compositions of linear transformations
3. The matrix of a linear transformation
4. Compositions ofmatrices
5. Change of basis. Equivalence and similarity of matrices
6. Rank space and null space of a linear transformation
7. Systems oflinear equations
8. Linear transformations in right vector spaces
9. Linear functions
10. Duality between a finite dimensional space and its .conjugate space
11. Transpose of a linear transformation
12. Matrices of the transpose
13. Projections
CHAPTER III: THE THEORY OF A SINGLE LINEAR TRANSFORMATION
1. The minimum polynomial of a linear transformation
2. Cyclicsubspaces
3. Existence of a vector whose order is the minimum polynomial
4. Cyclic linear transformations
5. The module det:ermiried by a linear transformation
6. Finitely generated o-modules, o, a principal ideal domain
7. Normalization of the generators of; and of
8. Equivalence of matrices with elements in a principal ideal domain
9. Structure of finitely generated o-modules
10. Invarjance theorems
11. Decomposition of a vector space relative to a linear trans- formation
12. The characteristic and minimum polynomials
13. Direct proof of Theorem 13
14. Formal properties of the trace and the characteristic poly- nomial
15. The ring of o-endomorphisms of a cyclic o-module
16. Determination of the ring of o-endomorphisms of a finitely generated o-module, o principal
17. The linear transformations which commute with a given lin- ear transformation
18. The center of the ring
CHAPTER Ⅳ: SETS OF LINEAR TRANSFORMATIONS
1. Invariant subspaces
2. Induced linear transformations
3. Composition series
4. Decomposability
5. Complete reducibility
6. Relation to the theory of operator groups and the theory of modules
7. Reducibility, decomposability, complete reducibility for a single linear transformation
8. The primary components of a space relative to a linear trans- formation
9. Sets of commutative linear transformations
CHAPTER Ⅴ: BILINEAR FORMS
1. Bilinear forms
2. Matrices of a bilinear form
……
CHAPTER VI: EUCLIDEAN AND UNITARY SPACES
CHAPTER Ⅶ: PRODUCTS OF VECTOR SPApES
CHAPTER Ⅷ: THE RING OF LINEAR TRANSFORMATIONS
CHAPTER IX: INFINITE DIMENSIONAL VECTOR SPACES
Index
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