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Linear Algebra Done Right
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S. Axler
On this page
▸
1 · Vector Spaces
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2 · Finite-Dimensional Vector Spaces
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3 · Linear Maps
4 · Polynomials
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5 · Eigenvalues and Eigenvectors
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6 · Inner Product Spaces
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7 · Operators on Inner Product Spaces
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8 · Operators on Complex Vector Spaces
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9 · Multilinear Algebra and Determinants
Chapter 1
Vector Spaces
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Section 1A
R
n
\mathbb{R}^n
R
n
and
C
n
\mathbb{C}^n
C
n
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Section 1B
Definition of Vector Space
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Section 1C
Subspaces
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Chapter 2
Finite-Dimensional Vector Spaces
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Section 2A
Span and Linear Independence
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Section 2B
Bases
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Section 2C
Dimension
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Chapter 3
Linear Maps
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Section 3A
Vector Space of Linear Maps
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Section 3B
Null Spaces and Ranges
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Section 3C
Matrices
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Section 3D
Invertibility and Isomorphisms
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Section 3E
Products and Quotients of Vector Spaces
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Section 3F
Duality
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Chapter 4
Polynomials
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Chapter 5
Eigenvalues and Eigenvectors
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Section 5A
Invariant Subspaces
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Section 5B
The Minimal Polynomial
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Section 5C
Upper-Triangular Matrices
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Section 5D
Diagonalizable Operators
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Section 5E
Commuting Operators
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Chapter 6
Inner Product Spaces
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Section 6A
Inner Products and Norms
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Section 6B
Orthonormal Bases
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Section 6C
Orthogonal Complements and Minimization Problems
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Chapter 7
Operators on Inner Product Spaces
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Section 7A
Self-Adjoint and Normal Operators
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Section 7B
Spectral Theorem
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Section 7C
Positive Operators
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Section 7D
Isometries, Unitary Operators, and Matrix Factorization
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Section 7E
Singular Value Decomposition
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Section 7F
Consequences of Singular Value Decomposition
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Chapter 8
Operators on Complex Vector Spaces
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Section 8A
Generalized Eigenvectors and Nilpotent Operators
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Section 8B
Generalized Eigenspace Decomposition
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Section 8C
Consequences of Generalized Eigenspace Decomposition
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Section 8D
Trace: a Connection between Matrices and Operators
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Chapter 9
Multilinear Algebra and Determinants
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Section 9A
Bilinear Forms and Quadratic Forms
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Section 9B
Alternating Multilinear Forms
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Section 9C
Determinants
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Section 9D
Tensor Products
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