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Introduction to algebra
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II · Linear algebra
Chapter II.1
Spaces and forms
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II.1.1 · Abstract vector spaces
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II.1.2 · Dimension and basis
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II.1.3 · Dual space
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II.1.4 · Bilinear and quadratic forms
Section II.1.1
Abstract vector spaces
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Subsection II.1.1.1
Motivation and axiomatization
Subsection II.1.1.2
Linear spans. Subspaces
Subsection II.1.1.3
Notes on the geometric interpretation
Section II.1.2
Dimension and basis
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Subsection II.1.2.1
Linear dependence
Subsection II.1.2.2
Dimension of a vector space and its basis
Subsection II.1.2.3
Coordinates. Isomorphism of spaces
Subsection II.1.2.4
Intersection and sum of subspaces
Subsection II.1.2.5
Direct sums
Subsection II.1.2.6
Quotient spaces
Section II.1.3
Dual space
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Subsection II.1.3.1
Linear functions
Subsection II.1.3.2
Dual space and dual basis
Subsection II.1.3.3
Reflexivity
Subsection II.1.3.4
Linear independence criterion
Subsection II.1.3.5
Geometric interpretation of solutions to a system of linear equations
Section II.1.4
Bilinear and quadratic forms
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Subsection II.1.4.1
Multilinear mappings
Subsection II.1.4.2
Bilinear forms
Subsection II.1.4.3
The law of change for the matrix of a bilinear form
Subsection II.1.4.4
Symmetric and skew-symmetric forms
Subsection II.1.4.5
Quadratic forms
Subsection II.1.4.6
The canonical form of a quadratic form
Subsection II.1.4.7
Real quadratic forms
Subsection II.1.4.8
Positive definite forms and matrices
Subsection II.1.4.9
The canonical form of a skew-symmetric form