# Book:Nathan Jacobson/Lectures in Abstract Algebra/Volume II

## Nathan Jacobson: Lectures in Abstract Algebra, Volume $\text { II }$: Linear Algebra

Published $\text {1953}$, Van Nostrand Reinhold

Linear Algebra

### Contents

Preface

Chapter $\text I$: Finite Dimensional Vector Spaces
1. Abstract vector spaces
2. Right vector spaces
3. $\mathfrak o$-modules
4. Linear dependence
5. Invariance of dimensionality
6. Bases and matrices
7. Applications to matrix theory
8. Rank of a set of vectors
9. Factor spaces
10. Algebra of subspaces
11. Independent subspaces, direct sums

Chapter $\text {II}$: Linear Transformations
1. Definition and examples
2. Composition of linear transformations
3. The matrix of a linear transformation
4. Composition of matrices
5. Change of basis. Equivalence and similarity of matrices
6. Rank space and null space of a linear transformation
7. Systems of linear equations
8. Linear transformations in right vector spaces
9. Linear functions
10. Duality between a finite dimensional linear space and its conjugate space
11. Transpose of a linear transformation
12. Matrices of the transpose
13. Projections

Chapter $\text {III}$: The Theory of a Single Linear Transformation
1. The minimum polynomial of a linear transformation
2. Cyclic subspaces
3. Existence of a vector whose order is the minimum polynomial
4. Cyclic linear transformations
5. The $\Phi \left[{\lambda}\right]$-module determined by a linear transformation
6. Finitely generated $\mathfrak o$-modules, $\mathfrak o$, a principal ideal domain
7. Normalization of the generators of $\mathfrak F$ and of $\mathfrak R$
8. Equivalence of matrices with elements in a principal ideal domain
9. Structure of finitely generated $\mathfrak o$-modules
10. Invariance theorems
11. Decomposition of a vector space relative to a linear transformation
12. The characteristic and minimum polynomials
13. Direct proof of Theorem 13
14. Formal properties of the trace and the characteristic polynomial
15. The ring of $\mathfrak o$-endomorphisms of a cyclic $\mathfrak o$-module
16. Determination of the ring of $\mathfrak o$-endomorphisms of a finitely generated $\mathfrak o$-module, $\mathfrak o$ principal
17. The linear transformations which commute with a given linear transformation
18. The center of the ring $\mathfrak B$

Chapter $\text {IV}$: 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 transformation
9. Sets of commutative linear transformations

Chapter $\text V$: Bilinear Forms
1. Bilinear forms
2. Matrices of a bilinear form
3. Non-degenerate forms
4. Transpose of a linear transformation relative to a pair of bilinear forms
5. Another relation between linear transformations and bilinear forms
6. Scalar products
7. Hermitian scalar products
8. Matrices of hermitian scalar products
9. Symmetric and hermitian scalar products over special division rings
10. Alternate scalar products
11. Witt's theorem
12. Non-alternate skew-symmetric forms

Chapter $\text {VI}$: Euclidean and Unitary Spaces
1. Cartesian bases
2. Linear transformations and scalar products
3. Orthogonal complete reducibility
4. Symmetric, skew and orthogonal linear transformations
5. Canonical matrices for symmetric and skew linear transformations
6. Commutative symmetric and skew linear transformations
7. Normal and orthogonal linear transformations
8. Semi-definite transformations
9. Polar factorization of an arbitrary linear transformation
10. Unitary geometry
11. Analytic functions of linear transformations

Chapter $\text {VII}$: Products of Vector Spaces
1. Product groups of vector spaces
2. Direct products of linear transformations
3. Two-sided vector spaces
4. The Kronecker product
5. Kronecker products of linear transformations and of matrices
6. Tensor spaces
7. Symmetry classes of tensors
8. Extension of the field of a vector space
9. A theorem on similarity of sets of matrices
10. Alternative definition of an algebra. Kronecker product of algebras

Chapter $\text {VIII}$: The Ring of Linear Transformations
1. Simplicity of $\mathfrak L$
2. Operator methods
3. The left ideals of $\mathfrak L$
4. Right ideals
5. Isomorphisms of rings of linear transformations

Chapter $\text {IX}$: Infinite Dimensional Vector Spaces
1. Existence of a basis
2. Invariance of dimensionality
3. Subspaces
4. Linear transformations and matrices
5. Dimensionality of the conjugate space
6. Finite topology for linear transformations
7. Total subspaces of $\mathfrak R^*$
8. Dual subspaces. Kronecker products
9. Two-sided ideals in the ring of linear transformations
10. Dense rings of linear transformations
11. Isomorphism theorems
12. Anti-automorphisms and scalar products
13. Schur's lemma. A general density theorem
14. Irreducible algebras of linear transformations

Index