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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

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.