What is the point of going from R, R 2and R 3, which seem comfortable and βreal worldβ, to Rn ? Our world is, after all, three-dimensional, not n-dimensional !
Euclidean Space Inner Product Orthogonal Orthonormal Basis Isomorphic Coordinate-Free Versus Basis Linear Transformation Adjoint Matrix Representation Adjoint as Transpose Riesz Representation Matrix Representation Fundamental Theorem of Linear Algebra Moore-Penrose Generalized Inverse Linear Equation Order Definition of the Inner Product To embed model structure into the inner product simplifies theanalysis.In a Euclidean space of random variables, one might define theinner product of two random variables as the covariance.Orthogonality then means no correlation.A different definition of the inner product derives from a partial ordering : one defines a βtraceβ inner product consistent withthe ordering.
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What is the point of going from R, R 2and R 3, which seem comfortable and βreal worldβ, to Rn ? Our world is, after all, three-dimensional, not n-dimensional !
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Euclidean Space
Inner Product
Orthogonal
Orthonormal Basis
Isomorphic
Coordinate-Free Versus Basis
Linear Transformation
Adjoint
Matrix Representation
Adjoint as Transpose
Riesz Representation
Matrix Representation
Fundamental Theorem of Linear Algebra
Moore-Penrose Generalized Inverse
Linear Equation
Order
Definition of the Inner Product
To embed model structure into the inner product simplifies theanalysis.In a Euclidean space of random variables, one might define theinner product of two random variables as the covariance.Orthogonality then means no correlation.A different definition of the inner product derives from a partial ordering : one defines a βtraceβ inner product consistent withthe ordering.
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