### Mathematical methods Lecture 12 of 34

October 11, 2012 by Multimedia Publications and Printing Services

Lecturer: K.S. Narain , ICTP

We continue with the summary of the rest of the course. In the 3rd and 4th lectures map between two vectors , Cauchy-Schwarz inequality and Orthogonalization Theorem was done. The existence of unique map between two vectors and definition of linear map is given. (given all properties) , Cauchy-Schwarz inequality and complex vector and their properties with some examples are introduced. The notion of distance between two vectors mentioned. Orthogonalization Theorem and Graham Schmidt method by giving several examples are realized. Transpose vector and its properties and Unitary group, unitary transformation ,parity transformation Definition of linear generator are explained. Linear operators and finite dimensional vector space with n dimension mentioned. Any finite vector space can be presented by linear operator with N*N matrix. Finite dimensional space and giving several examples for those matrices have been applied. Power series expansion and some application are given. Identity operator and its properties and the prove for existing of Inverse matrix are mentioned. The definition of adjoint operator and some of its properties have been said. Hermitian operator ,Projection operator , the definition of eigen value and eigen vectors are given. The prove of rank one for vectors is included .Ordinary eigenvector is also introduced. Generalized ordinary vector is also expressed

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