Linear Algebra by Serge Lang: Undergrad Theory & Applications

In summary, "Linear Algebra" by Serge Lang is a comprehensive textbook that covers the fundamentals of linear algebra. It assumes some familiarity with matrices and proofs and is suitable for undergraduate students. The book covers topics such as vectors, vector spaces, matrices, linear mappings, scalar products and orthogonality, determinants, structure theorems, and their relations with other structures such as multilinear products, groups, and rings. It also includes appendices on convex sets, induction, algebraic closure of complex numbers, equivalence relations, and angles.

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Table of Contents:
Code:
[LIST]
[*] Basic Theory
[LIST]
[*] Vectors
[LIST]
[*] Definition of points in n-space 
[*] Located vectors
[*] Scalar product 
[*] The norm of a vector 
[*] Lines and planes 
[*] The cross product 
[*] Complex numbers 
[/LIST]
[*] Vector Space
[LIST]
[*] Definitions 
[*] Bases
[*] Dimension of a vector space 
[*] Sums and direct sums
[/LIST]
[*] Matrices
[LIST]
[*] The space of matrices
[*] Linear equations
[*] Multiplication of matrices
[*] Appendix. Elimination
[/LIST]
[*] Linear Mappings 
[LIST]
[*] Mappings 
[*] Linear mappings 
[*] The kernel and image of a linear map 
[*] Composition and inverse of linear mappings. 
[*] Geometric applications
[/LIST]
[*] Linear Maps and Matrices 
[LIST]
[*] The linear map associated with a matrix 
[*] The matrix associated with a linear map
[*] Bases, matrices, and linear maps
[/LIST]
[*] Scalar Products and Orthogonality
[LIST]
[*] Scalar products
[*] Orthogonal bases, positive definite case
[*] Application to linear equations
[*] Bilinear maps and matrices
[*] General orthogonal bases
[*] The dual space
[/LIST]
[*] Determinants
[LIST]
[*] Determinants of order 2
[*] Existence of determinants
[*] Additional properties of determinants
[*] Cramer's rule
[*] Permutations 
[*] Uniqueness 
[*] Determinant of a transpose 
[*] Determinant of a product
[*] Inverse of a matrix
[*] The rank of a matrix and subdeterminants
[*] Determinants as area and volume 
[/LIST]
[/LIST]
[*] Structure Theorems 
[LIST]
[*] Bilinear Forms and the Standard Operators
[LIST]
[*] Bilinear forms 
[*] Quadratic forms
[*] Symmetric operators 
[*] Hermitian operators 
[*] Unitary operators 
[*] Sylvester's theorem
[/LIST]
[*] Polynomials and Matrices 
[LIST]
[*] Polynomials
[*] Polynomials of matrices and linear maps 
[*] Eigenvectors and eigenvalues
[*] The characteristic polynomial
[/LIST]
[*] Triangulation of Matrices and Linear Maps 
[LIST]
[*] Existence of triangulation
[*] Theorem of Hamilton-Cayley 
[*] Diagonalization of unitary maps
[/LIST]
[*] Spectral Theorem 
[LIST]
[*] Eigenvectors of symmetric linear maps
[*] The spectral theorem 
[*] The complex case 
[*] Unitary operators
[/LIST]
[*] Polynomials and Primary Decomposition
[LIST]
[*] The Euclidean algorithm 
[*] Greatest common divisor 
[*] Unique factorization
[*] The integers
[*] Application to the decomposition of a vector space
[*] Schur's lemma 
[*] The Jordan normal form 
[/LIST]
[/LIST]
[*] Relations with Other Structures 
[LIST]
[*] Multilinear Products 
[LIST]
[*] The tensor product
[*] Isomorphisms of tensor products
[*] Alternating products: Special case
[*] Alternating products: General case
[*] Appendix: The vector space generated by a set
[/LIST]
[*] Groups
[LIST]
[*] Groups and examples
[*] Simple properties of groups
[*] Cosets and normal subgroups
[*] Cyclic groups
[*] Free abelian groups
[/LIST]
[*] Rings
[LIST]
[*] Rings and ideals
[*] Homomorphisms
[*] Modules
[*] Factor modules
[/LIST]
[/LIST]
[*] Appendix: Convex Sets
[LIST]
[*]  Definitions 
[*] Separating hyperplanes
[*] Extreme points and supporting hyperplanes
[*] The Krein-Milman theorem 
[/LIST]
[*] Appendix: Odds and Ends 
[LIST]
[*] Induction 
[*] Algebraic closure of the complex numbers 
[*] Equivalence relations 
[/LIST]
[*] Appendix: Angles
[*] Answers 
[*] Index 
[/LIST]
 
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FAQ: Linear Algebra by Serge Lang: Undergrad Theory & Applications

1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with linear equations, vector spaces, and linear transformations. It involves the study of matrices, determinants, and systems of linear equations, and their properties and applications.

2. What are the main topics covered in "Linear Algebra by Serge Lang: Undergrad Theory & Applications"?

The main topics covered in this book include vector spaces, linear transformations, matrices, determinants, eigenvalues and eigenvectors, inner product spaces, and applications of linear algebra in various fields such as physics, engineering, and computer science.

3. Who is Serge Lang and why is his book on Linear Algebra important?

Serge Lang was a renowned mathematician and professor who made significant contributions to the field of mathematics. His book on Linear Algebra is important because it provides a rigorous and comprehensive treatment of the subject, making it suitable for undergraduate students and anyone interested in learning the theory and applications of linear algebra.

4. What are some common applications of Linear Algebra?

Linear Algebra has numerous applications in various fields such as physics, engineering, computer science, and economics. Some common applications include solving systems of linear equations, image and signal processing, data compression, and machine learning.

5. Is prior knowledge of Calculus required to understand "Linear Algebra by Serge Lang: Undergrad Theory & Applications"?

No, prior knowledge of Calculus is not required to understand this book. However, a basic understanding of algebra and functions is recommended. The book also includes a review of necessary mathematical concepts in the first chapter.

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