# ProofWiki:Books/Nathan Jacobson/Lectures in Abstract Algebra: II. Linear Algebra

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## Nathan Jacobson: *Lectures in Abstract Algebra: II. Linear Algebra*

Published 1953, **Van Nostrand Reinhold**.

### Subject Matter

### Contents

- Preface

- Chapter 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 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 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 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 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 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 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 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 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