By Sharipov R.
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Extra resources for Course of analytical geometry
Cn . § 12. LINEAR DEPENDENCE FOR n = 1. 1 are relatively simple. Their proofs are purely algebraic, they do not require drawings. I do not prove them in this book since the reader can find their proofs in § 3 of Chapter I in the book . 1, there is one more property which is formulated separately. 2 (Steinitz). If the vectors a1 , . . , an are linearly independent and if each of them is linearly expressed through some other vectors b1 , . . , bm , then m n. 2 is very important in studying multidimensional spaces.
The geometric construction shown in Fig. 1 can be applied to an arbitrary vector x. 6) of this vector in the basis e1 , e2 , e3 . However, this construction is not a unique way foe expanding a vector in a basis. For example, instead of the plane OAB the plane OAC can be taken for α, while the line can be directed parallel to the vector e2 . The the line EF is directed parallel to the vector e3 and the point F is obtained in its intersection with the line OA comprising the geometric realization of the vector e1 .
1. 7. 8. 7 can enter as subexpressions into simple and multiple sums which will be external sums with respect to them. Then some of their free indices or all of their free indices can turn into summation indices. Those of free indices that remain free are included into the list of free indices of the whole expression. 7 is applied. Taking into account this rue, each free index of the ultimate expression should enter it exactly once, while each summation index should enter it exactly twice — once as an upper index and once as a lower index.
Course of analytical geometry by Sharipov R.