Introduction
Inner Product Spaces
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Norms for Rn (17:01)
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Distance Between Vectors in Rn (9:20)
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Angle Between Vectors in Rn (19:57)
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Problem Set: Norms and Dot Product for Rn
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Inner Product Spaces (11:08)
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Examples of Inner Product (12:34)
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Norm, Distance, and Angle for Inner Product Spaces (18:56)
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Additional Example of Norm, Distance, and Angle for Inner Product Spaces (8:06)
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Orthogonal Projection (12:51)
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Problem Set: Inner Product Spaces
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Orthonormal Bases (10:55)
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Coordinates Relative to an Orthonormal Basis (15:15)
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Gram-Schmidt Process (8:09)
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Example of Gram-Schmidt Process (12:45)
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Additional Example of the Gram-Schmidt Process (15:42)
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Problem Set: Orthonormal Bases
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Least-Squares Problems (9:12)
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Example of Least-Squares Problem (13:59)
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Problem Set: Least Squares Problems
Linear Transformations
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Linear Transformations (14:09)
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Linear Transformations Represented by Matrices (6:55)
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Problem Set: Linear Transformations
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Kernel of a Linear Transformation (13:05)
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The Kernel of T as a Subspace of V (8:23)
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The Range of a Linear Transformation (5:41)
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Finding a Basis for Range(T) (4:58)
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Rank and Nullity of a Linear Transformation (9:47)
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One-to-one and Onto Properties (14:34)
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Isomorphisms (3:52)
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Problem Set: The Kernel and Range of a Linear Transformation
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Matrix Representation of Linear Transformations (14:52)
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Example of the Matrix of T Relative to the Bases B and B' (17:11)
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Additional Example of the Matrix of T Relative to the Bases B and B' (12:25)
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Problem Set: Matrix Representation of Linear Transformations
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Applications of Linear Transformations (8:04)
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Composition of Linear Transformations (5:48)
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Application to Computer Graphics (15:33)
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Problem Set: Applications of Linear Transformations
Eigenvalues and Eigenvectors
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Eigenvalues and Eigenvectors (14:37)
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Finding Eigenvalues and Eigenvectors of a Matrix (7:15)
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Example of Finding Eigenvalues and Eigenvectors (10:13)
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Additional Example of Finding Eigenvalues and Eigenvectors (14:36)
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Problem Set: Eigenvalues and Eigenvectors
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Diagonalization (13:01)
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A Necessary and Sufficient Condition for Diagonalizability (15:30)
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A Necessary and Sufficient Condition for Diagonalizability (Continued) (5:15)
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Diagonalizing a Matrix (4:43)
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Problem Set: Diagonalization
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Applications to Differential Equations (14:19)
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Example of Solving a System of Linear Differential Equations (13:38)
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Problem Set: Applications to Differential Equations
Symmetric Matrices and Orthogonal Diagonalization
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Symmetric Matrices (3:17)
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Example of Verifying Properties of Symmetric Matrices (17:52)
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Orthogonal Diagonalization (19:54)
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Summary of Orthogonal Diagonalization (6:45)
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The Spectral Theorem for Symmetric Matrices (3:31)
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Problem Set: Symmetric Matrices and Orthogonal Diagonalization
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Quadratic Forms (13:07)
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Eliminating Cross-product Terms (6:30)
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Example of Eliminating Cross-product Terms (13:41)
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The Principal Axes Theorem (5:37)
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Problem Set: Quadratic Forms
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Singular Value Decomposition (8:44)
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Example of Finding a Singular Value Decomposition of a Matrix (14:05)
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Example of Finding a Singular Value Decomposition of a Matrix (Continued) (13:50)
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Problem Set: Singular Value Decomposition
Application of SVD to Statistics: Principal Component Analysis
Conclusion
