GateXAIMLVector Spaces & Linear Algebra
  • Chapter 1: Vector Spaces
    • Vectors Concept
      • What is a Vector?
      • Geometric picture and basic diagrams
      • Length or Magnitude of a Vector
      • Vector arithmetic
      • Dot product (Inner product) and properties
      • Norms and distances
      • Cross product (3D) and properties
    • Vector Space
      • Concept of a Vector Space
      • Generalization to Higher Dimensions
      • Definition of a Real Vector Space
      • Eight Basic Axioms of Vector Spaces
    • Subspaces
    • Linear Independence, Basis, and Dimension
    • Applications of Vector Spaces
  • Chapter 2: Orthogonal Vectors, Subspaces, Matrices, and Projections
    • Orthogonal Vectors and Their Length
    • Orthogonal Subspaces and Complements
    • Orthogonal Matrices
    • Projection Matrices
    • Orthogonal and Symmetric Matrices
    • Applications of Orthogonality and Projections
  • Chapter 3: Quadratic Forms
    • Definition and Basic Properties
    • Diagonalization of Quadratic Forms
    • Positive Definite, Negative Definite, and Indefinite Forms
    • Canonical Form Using Orthogonal Transformation
    • Geometric Interpretation
    • Applications
  • Chapter 4: Singular Value Decomposition (SVD)
    • Definition and Concept
      • Reading the Axes: $U$, $\Sigma$, and $V$
      • Orthogonality
    • Eigenvalues, Eigenvectors, and the Origin of SVD
    • Computing the SVD
    • Low-Rank Approximation
    • Shortcut Rules for Special Matrices
    • Applications
GateXAIML · Linear Algebra Concept Notes

Vector Spaces & Linear Algebra

Chapter 1
Vector Spaces
16 sections
Vectors Concept, Vector Space, Subspaces...
Chapter 2
Orthogonal Vectors, Subspaces, Matrices, and Projections
6 sections
Orthogonal Vectors and Their Length, Orthogonal Subspaces and Complements, Orthogonal Matrices...
Chapter 3
Quadratic Forms
6 sections
Definition and Basic Properties, Diagonalization of Quadratic Forms, Positive Definite, Negative Definite, and Indefinite Forms...
Chapter 4
Singular Value Decomposition (SVD)
8 sections
Definition and Concept, Eigenvalues, Eigenvectors, and the Origin of SVD, Computing the SVD...
GateXAIML · Vector Spaces & Linear Algebra — Concept Notes
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