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

Subject Matter

 * 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