tailieunhanh - Handbook of mathematics for engineers and scienteists part 33

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 33', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 192 Algebra . Relations between coordinate transformations and basis transformations. Suppose that in a linear n-dimensional space V the transition from its basis ei . en to another basis ei . en is determined by the matrix A see Paragraph . Let x be any element of the space V with the coordinates xi . xn in the basis ei . en and the coordinates xi . Xn in the basis ei . en . x Xi ei x e xiei Xfitn. Then using formulas we obtain the following relations between these coordinates nn Xj Xiai3 Xk 2 Xibik j k i . n. In terms of matrices and row vectors these relations can be written as follows xi . Xn x 1 . Xn A xi . Xn xi . Xn A or in terms of column vectors Xi . . . Xn T AT Xi . . . Xn T Xi . . . Xn T A 1 T Xi . . . Xn T where the superscript T indicates the transpose of a matrix. . Euclidean Spaces . Real Euclidean Space . Definition and properties of a real Euclidean space. A real Euclidean space or simply Euclidean space is a real linear space V endowed with a scalar product also called inner product which is a real-valued function of two arguments x 6 V ye V called the scalar product of these elements denoted by x y and satisfying the following conditions axioms of the scalar product 1. Symmetry x y y x. 2. Distributivity xi x2 y xi y x2 y. 3. Homogeneity Ax y A x y for any real A. 4. Positive definiteness x x 0 for any x and x x 0 if and only if x 0. If the nature of the elements and the scalar product is concretized one obtains a specific Euclidean space. Example 1. Consider the linear space B3 of all free vectors in three-dimensional space. The space B3 becomes a Euclidean space if the scalar product is introduced as in analytic geometry see Paragraph x y x y cos where ip is the angle between the vectors x and y. Example 2. Consider the n-dimensional coordinate space Rn whose elements are ordered systems of n arbitrary real numbers x . Endowing this space with the scalar product x y xiyi xnyn we obtain a .