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By Sharipov R.

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In composing equalities the following rule is applied. 9. Both sides of an equality should have the same set of free indices and each free index should have the same position (upper or lower) in both sides of an equality. § 25. TRANSFORMATION OF THE COORDINATES . . 63 § 25. Transformation of the coordinates of vectors under a change of a basis. ˜1 , e ˜2 , e ˜3 be two bases in the space E and Let e1 , e2 , e3 and e ˜1 , e ˜2 , e ˜3 . As we already let e1 , e2 , e3 be changed for the basis e mentioned, in this case the basis e1 , e2 , e3 is called an old basis, ˜1 , e ˜2 , e ˜3 is called a new basis.

The geometric realizations of the vectors a1 , a2 , and a3 lie on the sides of this triangle. −−→ Therefore they lie on the plane ACE. Instead of a1 = AE , −−→ −−→ a2 = AB , and a3 = CD by means of parallel translations we can build some other geometric realizations of these three vectors. These geometric realizations do not lie on the plane ACE, but they keep parallelism to this plane. § 14. LINEAR DEPENDENCE FOR n = 3. 1. Three free vectors a1 , a2 , and a3 are called coplanar if their geometric realizations are parallel to some plane common to all three of them.

1. 3) is called the Kronecker symbol 1 or the Kronecker delta. The positions of indices in the Kronecker symbol are determined by a context where it is used. 1) can be written in components. In this particular case the indices of the Kronecker symbol are placed on different levels: 3 3 Sji Tkj = δki , Tji Skj = δki . 4) is inherited from the transition matrices S and T . Noter that the transition matrices S and T are square matrices. For such matrices the concept of the determinant is introduced (see [7]).

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