Orthogonal Diagonalization | Online Math Training

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Linear Algebra for Beginners: Open Doors to Great Careers 2

Introduction

Introduction Lecture (3:59)

Inner Product Spaces

Norms for Rn (17:01)
Distance Between Vectors in Rn (9:20)
Angle Between Vectors in Rn (19:57)
Problem Set: Norms and Dot Product for Rn
Inner Product Spaces (11:08)
Examples of Inner Product (12:34)
Norm, Distance, and Angle for Inner Product Spaces (18:56)
Additional Example of Norm, Distance, and Angle for Inner Product Spaces (8:06)
Orthogonal Projection (12:51)
Problem Set: Inner Product Spaces
Orthonormal Bases (10:55)
Coordinates Relative to an Orthonormal Basis (15:15)
Gram-Schmidt Process (8:09)
Example of Gram-Schmidt Process (12:45)
Additional Example of the Gram-Schmidt Process (15:42)
Problem Set: Orthonormal Bases
Least-Squares Problems (9:12)
Example of Least-Squares Problem (13:59)
Problem Set: Least Squares Problems

Linear Transformations

Linear Transformations (14:09)
Linear Transformations Represented by Matrices (6:55)
Problem Set: Linear Transformations
Kernel of a Linear Transformation (13:05)
The Kernel of T as a Subspace of V (8:23)
The Range of a Linear Transformation (5:41)
Finding a Basis for Range(T) (4:58)
Rank and Nullity of a Linear Transformation (9:47)
One-to-one and Onto Properties (14:34)
Isomorphisms (3:52)
Problem Set: The Kernel and Range of a Linear Transformation
Matrix Representation of Linear Transformations (14:52)
Example of the Matrix of T Relative to the Bases B and B' (17:11)
Additional Example of the Matrix of T Relative to the Bases B and B' (12:25)
Problem Set: Matrix Representation of Linear Transformations
Applications of Linear Transformations (8:04)
Composition of Linear Transformations (5:48)
Application to Computer Graphics (15:33)
Problem Set: Applications of Linear Transformations

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors (14:37)
Finding Eigenvalues and Eigenvectors of a Matrix (7:15)
Example of Finding Eigenvalues and Eigenvectors (10:13)
Additional Example of Finding Eigenvalues and Eigenvectors (14:36)
Problem Set: Eigenvalues and Eigenvectors
Diagonalization (13:01)
A Necessary and Sufficient Condition for Diagonalizability (15:30)
A Necessary and Sufficient Condition for Diagonalizability (Continued) (5:15)
Diagonalizing a Matrix (4:43)
Problem Set: Diagonalization
Applications to Differential Equations (14:19)
Example of Solving a System of Linear Differential Equations (13:38)
Problem Set: Applications to Differential Equations

Symmetric Matrices and Orthogonal Diagonalization

Symmetric Matrices (3:17)
Example of Verifying Properties of Symmetric Matrices (17:52)
Orthogonal Diagonalization (19:54)
Summary of Orthogonal Diagonalization (6:45)
The Spectral Theorem for Symmetric Matrices (3:31)
Problem Set: Symmetric Matrices and Orthogonal Diagonalization
Quadratic Forms (13:07)
Eliminating Cross-product Terms (6:30)
Example of Eliminating Cross-product Terms (13:41)
The Principal Axes Theorem (5:37)
Problem Set: Quadratic Forms
Singular Value Decomposition (8:44)
Example of Finding a Singular Value Decomposition of a Matrix (14:05)
Example of Finding a Singular Value Decomposition of a Matrix (Continued) (13:50)
Problem Set: Singular Value Decomposition

Application of SVD to Statistics: Principal Component Analysis

Mean-Deviation Form and Covariance Matrix (9:35)
Illustration of Mean-Deviation Form and Covariance Matrix (9:21)
Principal Component Analysis (10:16)

Conclusion

Concluding Letter

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Orthogonal Diagonalization

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