By Sharipov R.

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N. 5) j=1 i=1 that follows from the commutativity of the vector addition. 6) j=1 i=1 which is obtained by redesignating indices. 6) are used in dealing with vectors. 1 describes the property of the null vector. This property is often used in calculations. 3) is zero, e. g. if the equality n ai = 0 ak+1 + . . 3) can be transformed as follows: n a1 + . . + an = k ai = i=1 ai = a1 + . . + ak . 1 declares the existence of an opposite vector a′ for each vector a. Due to this item we can define the subtraction of vectors.

1. A system of three vectors a1 , a2 , a3 is linearly dependent if and only if these vectors are coplanar. § 15. Linear dependence for n 4. 1. Any system consisting of four vectors a1 , a2 , a3 , a4 in the space E is linearly dependent. , 2010. § 15. LINEAR DEPENDENCE FOR n 4. 2. Any system consisting of more than four vectors in the space E is linearly dependent. 1. 1. 1 itself expresses a property of the three-dimensional space E. 1. Let’s choose the subsystem composed by three vectors a1 , a2 , a3 within the system of four vectors a1 , a2 , a3 , a4 .

Each summation sign in a formula has its scope. This scope begins immediately after the summation sign to the right of it and ranges up to some delimiter: 1) the end of the formula; 60 CHAPTER I. VECTOR ALGEBRA. 2) the equality sign; 3) the plus sign «+» or the minus sign «−» not enclosed into brackets opened after a summation sign in question; 4) the closing bracket whose opening bracket precedes the summation sign in question. 3) and the comment to it). 2. Each summation index can be used only within the scope of the corresponding summation sign.