Linear Systems
 
Overview on linear systems
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Row Reduction
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Plan for Row Reduction
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Notation for Row Operations
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Algorithm for Row Reduction
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Matrices
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Matrix Operations and Matrix Algebra
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Matrix Equations
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The Superposition Principle
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Elementary Matrices
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Vector Spaces and Linear Transformations
 
Vector Spaces
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The vector space ℝn
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Definition of a vector space
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Subspaces
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Linear combinations and linear independence
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Constructing and Describing Vector Spaces and Subspaces
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Spanning sets, row spaces, and column spaces
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Nullspaces
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Range
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Bases and dimension
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Coordinate systems
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Vector spaces over ℂ
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Linear Transformations
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Definition
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Matrix representations of transformations
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Change of basis
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Vector spaces of linear transformations
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Eigenvalues and Eigenvectors
 
The Determinant
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Cofactor expansion
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Combinatorial definition
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Properties of the determinant
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Eigenvalues and Eigenvectors
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Definition
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Eigenspaces
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The characteristic polynomial
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Direct sum decomposition
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Properties of Eigenvalues and Eigenvectors
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Similarity and diagonalization
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Complex eigenvalues and eigenvectors
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Geometric versus algebraic multiplicity
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Shur’s Theorem
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Normal matrices
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Generalized eigenvectors
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Inner Product Spaces
 
Inner Products on ℝn
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The dot product in ℝn
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Symmetric bilinear pairings on ℝn
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Orthogonal vectors and subspaces in ℝn
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Orthonormal vectors and orthogonal matrices
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Projections and Least-squares Approximations
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Projection onto 1-dimensional subspaces
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Gram-Schmidt orthogonalization
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Least-squares approximations
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Least-squares solutions and the Fundamental Subspaces theorem
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Applications of least-squares solutions
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Projection onto a subspace
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Polynomial data fitting
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Complex inner product spaces
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The complex scalar product in ℂn
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Conjugate-symmetric sesquilinear pairings on ℂn, and their representation
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Unitary matrices
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Singular Values
 
Singular value decomposition
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Overall
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