Preface

1/19/2025

Introduction to Vector Spaces

1/19/2025

Definition and Examples

1/19/2025

Subspaces

1/19/2025

Span and Linear Independence

1/19/2025

Quotient Spaces

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Direct Sums

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Vector Space Homomorphisms

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Dual Spaces

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Basis and Dimension

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Basis of a Vector Space

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Dimension Theorem

1/19/2025

Finite and Infinite Dimensions

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Change of Basis

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Coordinate Systems

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Extension and Reduction of Bases

1/19/2025

Applications of Dimension

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Linear Transformations

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Kernel and Image

1/19/2025

Inner Product Spaces

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Orthogonality

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Gram-Schmidt Process

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Orthogonal Complements

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Orthonormal Bases

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Adjoint Operators

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Spectral Theorem

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Eigenvalues and Eigenvectors

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Characteristic Polynomial

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Diagonalization

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Eigenspaces

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The Cayley-Hamilton Theorem

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Jordan Canonical Form

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Applications in Differential Equations

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Spectral Decomposition

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Applications of Vector Spaces

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Differential Equations

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Fourier Series

1/19/2025

Quantum Mechanics

1/19/2025

Computer Graphics

1/19/2025

Coding Theory

1/19/2025

Economics and Game Theory

1/19/2025

Data Science

1/19/2025

Advanced Topics in Vector Spaces

1/19/2025

Tensor Products

1/19/2025

Exterior and Symmetric Algebras

1/19/2025

Modules over a Ring

1/19/2025

Topological Vector Spaces

1/19/2025

Normed and Banach Spaces

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Hilbert Spaces

1/19/2025

Representation Theory

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