By Sharipov R.

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Now let’s consider the inverse transition from the new basis ˜ ˜2 , e ˜3 to the old basis e1 , e2 , e3 . 6) ˜3 . 6) are called the inverse transition formulas. 6) are coordinates of the vectors e1 , e2 , e3 in their expansions in the new basis. These coefficients are arranged into columns: e1 → T31 T21 T11 e2 → T22 , T12 , T23 T13 e3 → T32 . 7) are united into a matrix: T = T11 T21 T31 T12 T22 T32 . 6. 6) is called the inverse transition matrix. 4): 3 ˜i , where j = 1, 2, 3. , 2010. 9) § 23. SOME INFORMATION .

Linear combinations. Triviality, non-triviality, and vanishing. Assume that some set of n free vectors a1 , . . , an is given. One can call it a collection of n vectors, a system of n vectors, or a family of n vectors either. Using the operation of vector addition and the operation of multiplication of vectors by numbers, one can compose some § 9. LINEAR COMBINATIONS. 33 vectorial expression of the vectors a1 , . . , an . It is quite likely that this expression will comprise sums of vectors taken with some numeric coefficients.

An . There are five basic properties of the linear dependence of vectors. We formulate them as a theorem. 1. The relation of the linear dependence for a system of vectors possesses the following basic properties: 1) a system of vectors comprising the null vector is linearly dependent; 2) a system of vectors comprising a linearly dependent subsystem is linearly dependent itself; 3) if a system of vectors is linearly dependent, then at least one of these vectors is expressed in a linear way through other vectors of this system; 4) if a system of vectors a1 , .