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The book devoted to matrix theory, linear systems and its properties, linear spaces, linear spaces with inner products, orthogonal systems, linear transforms in finite dimensional linear spaces, matrix of linear trasformations, symmetric and selfedjoint, positive transforms, diagonalization of matrix, Linear Transformation and Properties, Nullspace, Rank and Inverse of a Linear Transformation, Matrix Multiplication Transformations, Eigenvalues and Eigenvectors of a Linear Transformation, Conjugate Transformation, Selfadjoint Linear Transformations, Unitary Transformations, Normal Transformations, Positive Transformations, Orthogonal, Transformations Cayley Transformation, Normal and Jordan forms of transformations.