Linear Algebra by Serge Lang: Undergrad Theory & Applications

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• Author: Serge Lang
• Title: Linear Algebra
• Amazon Link: https://www.amazon.com/dp/1441930817/?tag=pfamazon01-20
• Prerequisities: Some familiarity with matrices and proofs
• Level: Undergrad

Table of Contents:

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[*] Basic Theory
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[*] Vectors
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[*] Definition of points in n-space 
[*] Located vectors
[*] Scalar product 
[*] The norm of a vector 
[*] Lines and planes 
[*] The cross product 
[*] Complex numbers 
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[*] Vector Space
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[*] Definitions 
[*] Bases
[*] Dimension of a vector space 
[*] Sums and direct sums
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[*] Matrices
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[*] The space of matrices
[*] Linear equations
[*] Multiplication of matrices
[*] Appendix. Elimination
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[*] Linear Mappings 
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[*] Mappings 
[*] Linear mappings 
[*] The kernel and image of a linear map 
[*] Composition and inverse of linear mappings. 
[*] Geometric applications
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[*] Linear Maps and Matrices 
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[*] The linear map associated with a matrix 
[*] The matrix associated with a linear map
[*] Bases, matrices, and linear maps
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[*] Scalar Products and Orthogonality
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[*] Scalar products
[*] Orthogonal bases, positive definite case
[*] Application to linear equations
[*] Bilinear maps and matrices
[*] General orthogonal bases
[*] The dual space
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[*] Determinants
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[*] 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 
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[*] Structure Theorems 
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[*] Bilinear Forms and the Standard Operators
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[*] Bilinear forms 
[*] Quadratic forms
[*] Symmetric operators 
[*] Hermitian operators 
[*] Unitary operators 
[*] Sylvester's theorem
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[*] Polynomials and Matrices 
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[*] Polynomials
[*] Polynomials of matrices and linear maps 
[*] Eigenvectors and eigenvalues
[*] The characteristic polynomial
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[*] Triangulation of Matrices and Linear Maps 
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[*] Existence of triangulation
[*] Theorem of Hamilton-Cayley 
[*] Diagonalization of unitary maps
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[*] Spectral Theorem 
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[*] Eigenvectors of symmetric linear maps
[*] The spectral theorem 
[*] The complex case 
[*] Unitary operators
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[*] Polynomials and Primary Decomposition
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[*] 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 
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[*] Relations with Other Structures 
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[*] Multilinear Products 
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[*] The tensor product
[*] Isomorphisms of tensor products
[*] Alternating products: Special case
[*] Alternating products: General case
[*] Appendix: The vector space generated by a set
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[*] Groups
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[*] Groups and examples
[*] Simple properties of groups
[*] Cosets and normal subgroups
[*] Cyclic groups
[*] Free abelian groups
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[*] Rings
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[*] Rings and ideals
[*] Homomorphisms
[*] Modules
[*] Factor modules
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[*] Appendix: Convex Sets
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[*]  Definitions 
[*] Separating hyperplanes
[*] Extreme points and supporting hyperplanes
[*] The Krein-Milman theorem 
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[*] Appendix: Odds and Ends 
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[*] Induction 
[*] Algebraic closure of the complex numbers 
[*] Equivalence relations 
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[*] Appendix: Angles
[*] Answers 
[*] Index 
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https://www.amazon.com/dp/1441930817/?tag=pfamazon01-20