By Sharipov R.

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**Extra info for Course of analytical geometry**

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5. If elements of a single index array are enumerated by upper indices, then in matrix presentation they are written in a column, i. e. they constitute a matrix whose width is equal to unity. 56 CHAPTER I. VECTOR ALGEBRA. 5. 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 .

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. 1. An expression of the form α1 a1 + . . + αn an composed of the vectors a1 , .

Let’s build −−→ −−→ the geometric realizations a2 = AB and β2 a2 = AC . Then at the point C we build the geometric realizations of the vectors −−→ −−→ −−→ −−→ a3 = CD and β3 a3 = CE . The vectors AC and CE constitute two sides of the triangle ACE (see Fig. 1). 1) is presented by the third side a1 = AE . The triangle ACE is a planar form. 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.