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Learn the core topics of Linear Algebra to open doors to Computer Science, Data Science, Actuarial Science, and more!

(1) Introduction

  • Introduction
  • Norms for Rn
  • Distance Between Vectors in Rn
  • Angle Between Vectors in Rn
  • Problem Set: Norms and Dot Product for Rn
  • Solutions to Problem Set: Norms and Dot Product for Rn
  • Inner Product Spaces
  • Examples of Inner Product
  • Norm, Distance, and Angle for Inner Product Spaces
  • Additional Example of Norm, Distance, and Angle for Inner Product Spaces
  • Orthogonal Projection
  • Problem Set: Inner Product Spaces
  • Solutions to Problem Set: Inner Product Spaces
  • Orthonormal Bases
  • Coordinates Relative to an Orthonormal Basis
  • Gram-Schmidt Process
  • Example of Gram-Schmidt Process
  • Additional Example of the Gram-Schmidt Process
  • Problem Set: Orthonormal Bases
  • Solutions to Problem Set: Orthonormal Bases
  • Least-Squares Problems
  • Example of Least-Squares Problem
  • Problem Set: Least Squares Problems
  • Solutions to Problem Set: Least Squares Problems
  • Linear Transformations
  • Linear Transformations Represented by Matrices
  • Problem Set: Linear Transformations
  • Solutions to Problem Set: Linear Transformations
  • Kernel of a Linear Transformation
  • The Kernel of T as a Subspace of V
  • The Range of a Linear Transformation
  • Finding a Basis for Range(T)
  • Rank and Nullity of a Linear Transformation
  • One-to-one and Onto Properties
  • Isomorphisms
  • Problem Set: The Kernel and Range of a Linear Transformation
  • Solutions to Problem Set: The Kernel and Range of a Linear Transformation
  • Matrix Representation of Linear Transformations
  • Example of the Matrix of T Relative to the Bases B and B'
  • Additional Example of the Matrix of T Relative to the Bases B and B'
  • Problem Set: Matrix Representation of Linear Transformations
  • Solutions to Problem Set: Matrix Representation of Linear Transformations
  • Applications of Linear Transformations
  • Composition of Linear Transformations
  • Application to Computer Graphics
  • Problem Set: Applications of Linear Transformations
  • Solutions to Problem Set: Applications of Linear Transformations
  • Review Request
  • Eigenvalues and Eigenvectors
  • Finding Eigenvalues and Eigenvectors of a Matrix
  • Example of Finding Eigenvalues and Eigenvectors
  • Additional Example of Finding Eigenvalues and Eigenvectors
  • Problem Set: Eigenvalues and Eigenvectors
  • Solutions to Problem Set: Eigenvalues and Eigenvectors
  • Diagonalization
  • A Necessary and Sufficient Condition for Diagonalizability
  • A Necessary and Sufficient Condition for Diagonalizability (Continued)
  • Diagonalizing a Matrix
  • Problem Set: Diagonalization
  • Solutions to Problem Set: Diagonalization
  • Applications to Differential Equations
  • Example of Solving a System of Linear Differential Equations
  • Problem Set: Applications to Differential Equations
  • Solutions to Problem Set: Applications to Differential Equations
  • Symmetric Matrices
  • Example of Verifying Properties of Symmetric Matrices
  • Orthogonal Diagonalization
  • Summary of Orthogonal Diagonalization
  • The Spectral Theorem for Symmetric Matrices
  • Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Solutions to Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Quadratic Forms
  • Eliminating Cross-product Terms
  • Example of Eliminating Cross-product Terms
  • The Principal Axes Theorem
  • Problem Set: Quadratic Forms
  • Solutions to Problem Set: Quadratic Forms
  • Singular Value Decomposition
  • Example of Finding a Singular Value Decomposition of a Matrix
  • Example of Finding a Singular Value Decomposition of a Matrix (Continued)
  • Problem Set: Singular Value Decomposition
  • Solutions to Problem Set: Singular Value Decomposition
  • Mean-Deviation Form and Covariance Matrix
  • Illustration of Mean-Deviation Form and Covariance Matrix
  • Principal Component Analysis
  • Concluding Letter
  • Bonus Lecture

(2) Inner Product Spaces

  • Introduction
  • Norms for Rn
  • Distance Between Vectors in Rn
  • Angle Between Vectors in Rn
  • Problem Set: Norms and Dot Product for Rn
  • Solutions to Problem Set: Norms and Dot Product for Rn
  • Inner Product Spaces
  • Examples of Inner Product
  • Norm, Distance, and Angle for Inner Product Spaces
  • Additional Example of Norm, Distance, and Angle for Inner Product Spaces
  • Orthogonal Projection
  • Problem Set: Inner Product Spaces
  • Solutions to Problem Set: Inner Product Spaces
  • Orthonormal Bases
  • Coordinates Relative to an Orthonormal Basis
  • Gram-Schmidt Process
  • Example of Gram-Schmidt Process
  • Additional Example of the Gram-Schmidt Process
  • Problem Set: Orthonormal Bases
  • Solutions to Problem Set: Orthonormal Bases
  • Least-Squares Problems
  • Example of Least-Squares Problem
  • Problem Set: Least Squares Problems
  • Solutions to Problem Set: Least Squares Problems
  • Linear Transformations
  • Linear Transformations Represented by Matrices
  • Problem Set: Linear Transformations
  • Solutions to Problem Set: Linear Transformations
  • Kernel of a Linear Transformation
  • The Kernel of T as a Subspace of V
  • The Range of a Linear Transformation
  • Finding a Basis for Range(T)
  • Rank and Nullity of a Linear Transformation
  • One-to-one and Onto Properties
  • Isomorphisms
  • Problem Set: The Kernel and Range of a Linear Transformation
  • Solutions to Problem Set: The Kernel and Range of a Linear Transformation
  • Matrix Representation of Linear Transformations
  • Example of the Matrix of T Relative to the Bases B and B'
  • Additional Example of the Matrix of T Relative to the Bases B and B'
  • Problem Set: Matrix Representation of Linear Transformations
  • Solutions to Problem Set: Matrix Representation of Linear Transformations
  • Applications of Linear Transformations
  • Composition of Linear Transformations
  • Application to Computer Graphics
  • Problem Set: Applications of Linear Transformations
  • Solutions to Problem Set: Applications of Linear Transformations
  • Review Request
  • Eigenvalues and Eigenvectors
  • Finding Eigenvalues and Eigenvectors of a Matrix
  • Example of Finding Eigenvalues and Eigenvectors
  • Additional Example of Finding Eigenvalues and Eigenvectors
  • Problem Set: Eigenvalues and Eigenvectors
  • Solutions to Problem Set: Eigenvalues and Eigenvectors
  • Diagonalization
  • A Necessary and Sufficient Condition for Diagonalizability
  • A Necessary and Sufficient Condition for Diagonalizability (Continued)
  • Diagonalizing a Matrix
  • Problem Set: Diagonalization
  • Solutions to Problem Set: Diagonalization
  • Applications to Differential Equations
  • Example of Solving a System of Linear Differential Equations
  • Problem Set: Applications to Differential Equations
  • Solutions to Problem Set: Applications to Differential Equations
  • Symmetric Matrices
  • Example of Verifying Properties of Symmetric Matrices
  • Orthogonal Diagonalization
  • Summary of Orthogonal Diagonalization
  • The Spectral Theorem for Symmetric Matrices
  • Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Solutions to Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Quadratic Forms
  • Eliminating Cross-product Terms
  • Example of Eliminating Cross-product Terms
  • The Principal Axes Theorem
  • Problem Set: Quadratic Forms
  • Solutions to Problem Set: Quadratic Forms
  • Singular Value Decomposition
  • Example of Finding a Singular Value Decomposition of a Matrix
  • Example of Finding a Singular Value Decomposition of a Matrix (Continued)
  • Problem Set: Singular Value Decomposition
  • Solutions to Problem Set: Singular Value Decomposition
  • Mean-Deviation Form and Covariance Matrix
  • Illustration of Mean-Deviation Form and Covariance Matrix
  • Principal Component Analysis
  • Concluding Letter
  • Bonus Lecture

(3) Linear Transformations

  • Introduction
  • Norms for Rn
  • Distance Between Vectors in Rn
  • Angle Between Vectors in Rn
  • Problem Set: Norms and Dot Product for Rn
  • Solutions to Problem Set: Norms and Dot Product for Rn
  • Inner Product Spaces
  • Examples of Inner Product
  • Norm, Distance, and Angle for Inner Product Spaces
  • Additional Example of Norm, Distance, and Angle for Inner Product Spaces
  • Orthogonal Projection
  • Problem Set: Inner Product Spaces
  • Solutions to Problem Set: Inner Product Spaces
  • Orthonormal Bases
  • Coordinates Relative to an Orthonormal Basis
  • Gram-Schmidt Process
  • Example of Gram-Schmidt Process
  • Additional Example of the Gram-Schmidt Process
  • Problem Set: Orthonormal Bases
  • Solutions to Problem Set: Orthonormal Bases
  • Least-Squares Problems
  • Example of Least-Squares Problem
  • Problem Set: Least Squares Problems
  • Solutions to Problem Set: Least Squares Problems
  • Linear Transformations
  • Linear Transformations Represented by Matrices
  • Problem Set: Linear Transformations
  • Solutions to Problem Set: Linear Transformations
  • Kernel of a Linear Transformation
  • The Kernel of T as a Subspace of V
  • The Range of a Linear Transformation
  • Finding a Basis for Range(T)
  • Rank and Nullity of a Linear Transformation
  • One-to-one and Onto Properties
  • Isomorphisms
  • Problem Set: The Kernel and Range of a Linear Transformation
  • Solutions to Problem Set: The Kernel and Range of a Linear Transformation
  • Matrix Representation of Linear Transformations
  • Example of the Matrix of T Relative to the Bases B and B'
  • Additional Example of the Matrix of T Relative to the Bases B and B'
  • Problem Set: Matrix Representation of Linear Transformations
  • Solutions to Problem Set: Matrix Representation of Linear Transformations
  • Applications of Linear Transformations
  • Composition of Linear Transformations
  • Application to Computer Graphics
  • Problem Set: Applications of Linear Transformations
  • Solutions to Problem Set: Applications of Linear Transformations
  • Review Request
  • Eigenvalues and Eigenvectors
  • Finding Eigenvalues and Eigenvectors of a Matrix
  • Example of Finding Eigenvalues and Eigenvectors
  • Additional Example of Finding Eigenvalues and Eigenvectors
  • Problem Set: Eigenvalues and Eigenvectors
  • Solutions to Problem Set: Eigenvalues and Eigenvectors
  • Diagonalization
  • A Necessary and Sufficient Condition for Diagonalizability
  • A Necessary and Sufficient Condition for Diagonalizability (Continued)
  • Diagonalizing a Matrix
  • Problem Set: Diagonalization
  • Solutions to Problem Set: Diagonalization
  • Applications to Differential Equations
  • Example of Solving a System of Linear Differential Equations
  • Problem Set: Applications to Differential Equations
  • Solutions to Problem Set: Applications to Differential Equations
  • Symmetric Matrices
  • Example of Verifying Properties of Symmetric Matrices
  • Orthogonal Diagonalization
  • Summary of Orthogonal Diagonalization
  • The Spectral Theorem for Symmetric Matrices
  • Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Solutions to Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Quadratic Forms
  • Eliminating Cross-product Terms
  • Example of Eliminating Cross-product Terms
  • The Principal Axes Theorem
  • Problem Set: Quadratic Forms
  • Solutions to Problem Set: Quadratic Forms
  • Singular Value Decomposition
  • Example of Finding a Singular Value Decomposition of a Matrix
  • Example of Finding a Singular Value Decomposition of a Matrix (Continued)
  • Problem Set: Singular Value Decomposition
  • Solutions to Problem Set: Singular Value Decomposition
  • Mean-Deviation Form and Covariance Matrix
  • Illustration of Mean-Deviation Form and Covariance Matrix
  • Principal Component Analysis
  • Concluding Letter
  • Bonus Lecture

(4) Eigenvalues and Eigenvectors

  • Introduction
  • Norms for Rn
  • Distance Between Vectors in Rn
  • Angle Between Vectors in Rn
  • Problem Set: Norms and Dot Product for Rn
  • Solutions to Problem Set: Norms and Dot Product for Rn
  • Inner Product Spaces
  • Examples of Inner Product
  • Norm, Distance, and Angle for Inner Product Spaces
  • Additional Example of Norm, Distance, and Angle for Inner Product Spaces
  • Orthogonal Projection
  • Problem Set: Inner Product Spaces
  • Solutions to Problem Set: Inner Product Spaces
  • Orthonormal Bases
  • Coordinates Relative to an Orthonormal Basis
  • Gram-Schmidt Process
  • Example of Gram-Schmidt Process
  • Additional Example of the Gram-Schmidt Process
  • Problem Set: Orthonormal Bases
  • Solutions to Problem Set: Orthonormal Bases
  • Least-Squares Problems
  • Example of Least-Squares Problem
  • Problem Set: Least Squares Problems
  • Solutions to Problem Set: Least Squares Problems
  • Linear Transformations
  • Linear Transformations Represented by Matrices
  • Problem Set: Linear Transformations
  • Solutions to Problem Set: Linear Transformations
  • Kernel of a Linear Transformation
  • The Kernel of T as a Subspace of V
  • The Range of a Linear Transformation
  • Finding a Basis for Range(T)
  • Rank and Nullity of a Linear Transformation
  • One-to-one and Onto Properties
  • Isomorphisms
  • Problem Set: The Kernel and Range of a Linear Transformation
  • Solutions to Problem Set: The Kernel and Range of a Linear Transformation
  • Matrix Representation of Linear Transformations
  • Example of the Matrix of T Relative to the Bases B and B'
  • Additional Example of the Matrix of T Relative to the Bases B and B'
  • Problem Set: Matrix Representation of Linear Transformations
  • Solutions to Problem Set: Matrix Representation of Linear Transformations
  • Applications of Linear Transformations
  • Composition of Linear Transformations
  • Application to Computer Graphics
  • Problem Set: Applications of Linear Transformations
  • Solutions to Problem Set: Applications of Linear Transformations
  • Review Request
  • Eigenvalues and Eigenvectors
  • Finding Eigenvalues and Eigenvectors of a Matrix
  • Example of Finding Eigenvalues and Eigenvectors
  • Additional Example of Finding Eigenvalues and Eigenvectors
  • Problem Set: Eigenvalues and Eigenvectors
  • Solutions to Problem Set: Eigenvalues and Eigenvectors
  • Diagonalization
  • A Necessary and Sufficient Condition for Diagonalizability
  • A Necessary and Sufficient Condition for Diagonalizability (Continued)
  • Diagonalizing a Matrix
  • Problem Set: Diagonalization
  • Solutions to Problem Set: Diagonalization
  • Applications to Differential Equations
  • Example of Solving a System of Linear Differential Equations
  • Problem Set: Applications to Differential Equations
  • Solutions to Problem Set: Applications to Differential Equations
  • Symmetric Matrices
  • Example of Verifying Properties of Symmetric Matrices
  • Orthogonal Diagonalization
  • Summary of Orthogonal Diagonalization
  • The Spectral Theorem for Symmetric Matrices
  • Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Solutions to Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Quadratic Forms
  • Eliminating Cross-product Terms
  • Example of Eliminating Cross-product Terms
  • The Principal Axes Theorem
  • Problem Set: Quadratic Forms
  • Solutions to Problem Set: Quadratic Forms
  • Singular Value Decomposition
  • Example of Finding a Singular Value Decomposition of a Matrix
  • Example of Finding a Singular Value Decomposition of a Matrix (Continued)
  • Problem Set: Singular Value Decomposition
  • Solutions to Problem Set: Singular Value Decomposition
  • Mean-Deviation Form and Covariance Matrix
  • Illustration of Mean-Deviation Form and Covariance Matrix
  • Principal Component Analysis
  • Concluding Letter
  • Bonus Lecture

(5) Symmetric Matrices and Orthogonal Diagonalization

  • Introduction
  • Norms for Rn
  • Distance Between Vectors in Rn
  • Angle Between Vectors in Rn
  • Problem Set: Norms and Dot Product for Rn
  • Solutions to Problem Set: Norms and Dot Product for Rn
  • Inner Product Spaces
  • Examples of Inner Product
  • Norm, Distance, and Angle for Inner Product Spaces
  • Additional Example of Norm, Distance, and Angle for Inner Product Spaces
  • Orthogonal Projection
  • Problem Set: Inner Product Spaces
  • Solutions to Problem Set: Inner Product Spaces
  • Orthonormal Bases
  • Coordinates Relative to an Orthonormal Basis
  • Gram-Schmidt Process
  • Example of Gram-Schmidt Process
  • Additional Example of the Gram-Schmidt Process
  • Problem Set: Orthonormal Bases
  • Solutions to Problem Set: Orthonormal Bases
  • Least-Squares Problems
  • Example of Least-Squares Problem
  • Problem Set: Least Squares Problems
  • Solutions to Problem Set: Least Squares Problems
  • Linear Transformations
  • Linear Transformations Represented by Matrices
  • Problem Set: Linear Transformations
  • Solutions to Problem Set: Linear Transformations
  • Kernel of a Linear Transformation
  • The Kernel of T as a Subspace of V
  • The Range of a Linear Transformation
  • Finding a Basis for Range(T)
  • Rank and Nullity of a Linear Transformation
  • One-to-one and Onto Properties
  • Isomorphisms
  • Problem Set: The Kernel and Range of a Linear Transformation
  • Solutions to Problem Set: The Kernel and Range of a Linear Transformation
  • Matrix Representation of Linear Transformations
  • Example of the Matrix of T Relative to the Bases B and B'
  • Additional Example of the Matrix of T Relative to the Bases B and B'
  • Problem Set: Matrix Representation of Linear Transformations
  • Solutions to Problem Set: Matrix Representation of Linear Transformations
  • Applications of Linear Transformations
  • Composition of Linear Transformations
  • Application to Computer Graphics
  • Problem Set: Applications of Linear Transformations
  • Solutions to Problem Set: Applications of Linear Transformations
  • Review Request
  • Eigenvalues and Eigenvectors
  • Finding Eigenvalues and Eigenvectors of a Matrix
  • Example of Finding Eigenvalues and Eigenvectors
  • Additional Example of Finding Eigenvalues and Eigenvectors
  • Problem Set: Eigenvalues and Eigenvectors
  • Solutions to Problem Set: Eigenvalues and Eigenvectors
  • Diagonalization
  • A Necessary and Sufficient Condition for Diagonalizability
  • A Necessary and Sufficient Condition for Diagonalizability (Continued)
  • Diagonalizing a Matrix
  • Problem Set: Diagonalization
  • Solutions to Problem Set: Diagonalization
  • Applications to Differential Equations
  • Example of Solving a System of Linear Differential Equations
  • Problem Set: Applications to Differential Equations
  • Solutions to Problem Set: Applications to Differential Equations
  • Symmetric Matrices
  • Example of Verifying Properties of Symmetric Matrices
  • Orthogonal Diagonalization
  • Summary of Orthogonal Diagonalization
  • The Spectral Theorem for Symmetric Matrices
  • Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Solutions to Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Quadratic Forms
  • Eliminating Cross-product Terms
  • Example of Eliminating Cross-product Terms
  • The Principal Axes Theorem
  • Problem Set: Quadratic Forms
  • Solutions to Problem Set: Quadratic Forms
  • Singular Value Decomposition
  • Example of Finding a Singular Value Decomposition of a Matrix
  • Example of Finding a Singular Value Decomposition of a Matrix (Continued)
  • Problem Set: Singular Value Decomposition
  • Solutions to Problem Set: Singular Value Decomposition
  • Mean-Deviation Form and Covariance Matrix
  • Illustration of Mean-Deviation Form and Covariance Matrix
  • Principal Component Analysis
  • Concluding Letter
  • Bonus Lecture

(6) Application of SVD to Statistics: Principal Component Analysis

  • Introduction
  • Norms for Rn
  • Distance Between Vectors in Rn
  • Angle Between Vectors in Rn
  • Problem Set: Norms and Dot Product for Rn
  • Solutions to Problem Set: Norms and Dot Product for Rn
  • Inner Product Spaces
  • Examples of Inner Product
  • Norm, Distance, and Angle for Inner Product Spaces
  • Additional Example of Norm, Distance, and Angle for Inner Product Spaces
  • Orthogonal Projection
  • Problem Set: Inner Product Spaces
  • Solutions to Problem Set: Inner Product Spaces
  • Orthonormal Bases
  • Coordinates Relative to an Orthonormal Basis
  • Gram-Schmidt Process
  • Example of Gram-Schmidt Process
  • Additional Example of the Gram-Schmidt Process
  • Problem Set: Orthonormal Bases
  • Solutions to Problem Set: Orthonormal Bases
  • Least-Squares Problems
  • Example of Least-Squares Problem
  • Problem Set: Least Squares Problems
  • Solutions to Problem Set: Least Squares Problems
  • Linear Transformations
  • Linear Transformations Represented by Matrices
  • Problem Set: Linear Transformations
  • Solutions to Problem Set: Linear Transformations
  • Kernel of a Linear Transformation
  • The Kernel of T as a Subspace of V
  • The Range of a Linear Transformation
  • Finding a Basis for Range(T)
  • Rank and Nullity of a Linear Transformation
  • One-to-one and Onto Properties
  • Isomorphisms
  • Problem Set: The Kernel and Range of a Linear Transformation
  • Solutions to Problem Set: The Kernel and Range of a Linear Transformation
  • Matrix Representation of Linear Transformations
  • Example of the Matrix of T Relative to the Bases B and B'
  • Additional Example of the Matrix of T Relative to the Bases B and B'
  • Problem Set: Matrix Representation of Linear Transformations
  • Solutions to Problem Set: Matrix Representation of Linear Transformations
  • Applications of Linear Transformations
  • Composition of Linear Transformations
  • Application to Computer Graphics
  • Problem Set: Applications of Linear Transformations
  • Solutions to Problem Set: Applications of Linear Transformations
  • Review Request
  • Eigenvalues and Eigenvectors
  • Finding Eigenvalues and Eigenvectors of a Matrix
  • Example of Finding Eigenvalues and Eigenvectors
  • Additional Example of Finding Eigenvalues and Eigenvectors
  • Problem Set: Eigenvalues and Eigenvectors
  • Solutions to Problem Set: Eigenvalues and Eigenvectors
  • Diagonalization
  • A Necessary and Sufficient Condition for Diagonalizability
  • A Necessary and Sufficient Condition for Diagonalizability (Continued)
  • Diagonalizing a Matrix
  • Problem Set: Diagonalization
  • Solutions to Problem Set: Diagonalization
  • Applications to Differential Equations
  • Example of Solving a System of Linear Differential Equations
  • Problem Set: Applications to Differential Equations
  • Solutions to Problem Set: Applications to Differential Equations
  • Symmetric Matrices
  • Example of Verifying Properties of Symmetric Matrices
  • Orthogonal Diagonalization
  • Summary of Orthogonal Diagonalization
  • The Spectral Theorem for Symmetric Matrices
  • Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Solutions to Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Quadratic Forms
  • Eliminating Cross-product Terms
  • Example of Eliminating Cross-product Terms
  • The Principal Axes Theorem
  • Problem Set: Quadratic Forms
  • Solutions to Problem Set: Quadratic Forms
  • Singular Value Decomposition
  • Example of Finding a Singular Value Decomposition of a Matrix
  • Example of Finding a Singular Value Decomposition of a Matrix (Continued)
  • Problem Set: Singular Value Decomposition
  • Solutions to Problem Set: Singular Value Decomposition
  • Mean-Deviation Form and Covariance Matrix
  • Illustration of Mean-Deviation Form and Covariance Matrix
  • Principal Component Analysis
  • Concluding Letter
  • Bonus Lecture

(7) Conclusion

  • Introduction
  • Norms for Rn
  • Distance Between Vectors in Rn
  • Angle Between Vectors in Rn
  • Problem Set: Norms and Dot Product for Rn
  • Solutions to Problem Set: Norms and Dot Product for Rn
  • Inner Product Spaces
  • Examples of Inner Product
  • Norm, Distance, and Angle for Inner Product Spaces
  • Additional Example of Norm, Distance, and Angle for Inner Product Spaces
  • Orthogonal Projection
  • Problem Set: Inner Product Spaces
  • Solutions to Problem Set: Inner Product Spaces
  • Orthonormal Bases
  • Coordinates Relative to an Orthonormal Basis
  • Gram-Schmidt Process
  • Example of Gram-Schmidt Process
  • Additional Example of the Gram-Schmidt Process
  • Problem Set: Orthonormal Bases
  • Solutions to Problem Set: Orthonormal Bases
  • Least-Squares Problems
  • Example of Least-Squares Problem
  • Problem Set: Least Squares Problems
  • Solutions to Problem Set: Least Squares Problems
  • Linear Transformations
  • Linear Transformations Represented by Matrices
  • Problem Set: Linear Transformations
  • Solutions to Problem Set: Linear Transformations
  • Kernel of a Linear Transformation
  • The Kernel of T as a Subspace of V
  • The Range of a Linear Transformation
  • Finding a Basis for Range(T)
  • Rank and Nullity of a Linear Transformation
  • One-to-one and Onto Properties
  • Isomorphisms
  • Problem Set: The Kernel and Range of a Linear Transformation
  • Solutions to Problem Set: The Kernel and Range of a Linear Transformation
  • Matrix Representation of Linear Transformations
  • Example of the Matrix of T Relative to the Bases B and B'
  • Additional Example of the Matrix of T Relative to the Bases B and B'
  • Problem Set: Matrix Representation of Linear Transformations
  • Solutions to Problem Set: Matrix Representation of Linear Transformations
  • Applications of Linear Transformations
  • Composition of Linear Transformations
  • Application to Computer Graphics
  • Problem Set: Applications of Linear Transformations
  • Solutions to Problem Set: Applications of Linear Transformations
  • Review Request
  • Eigenvalues and Eigenvectors
  • Finding Eigenvalues and Eigenvectors of a Matrix
  • Example of Finding Eigenvalues and Eigenvectors
  • Additional Example of Finding Eigenvalues and Eigenvectors
  • Problem Set: Eigenvalues and Eigenvectors
  • Solutions to Problem Set: Eigenvalues and Eigenvectors
  • Diagonalization
  • A Necessary and Sufficient Condition for Diagonalizability
  • A Necessary and Sufficient Condition for Diagonalizability (Continued)
  • Diagonalizing a Matrix
  • Problem Set: Diagonalization
  • Solutions to Problem Set: Diagonalization
  • Applications to Differential Equations
  • Example of Solving a System of Linear Differential Equations
  • Problem Set: Applications to Differential Equations
  • Solutions to Problem Set: Applications to Differential Equations
  • Symmetric Matrices
  • Example of Verifying Properties of Symmetric Matrices
  • Orthogonal Diagonalization
  • Summary of Orthogonal Diagonalization
  • The Spectral Theorem for Symmetric Matrices
  • Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Solutions to Problem Set: Symmetric Matrices and Orthogonal Diagonalization
  • Quadratic Forms
  • Eliminating Cross-product Terms
  • Example of Eliminating Cross-product Terms
  • The Principal Axes Theorem
  • Problem Set: Quadratic Forms
  • Solutions to Problem Set: Quadratic Forms
  • Singular Value Decomposition
  • Example of Finding a Singular Value Decomposition of a Matrix
  • Example of Finding a Singular Value Decomposition of a Matrix (Continued)
  • Problem Set: Singular Value Decomposition
  • Solutions to Problem Set: Singular Value Decomposition
  • Mean-Deviation Form and Covariance Matrix
  • Illustration of Mean-Deviation Form and Covariance Matrix
  • Principal Component Analysis
  • Concluding Letter
  • Bonus Lecture

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