By Sharipov R.A.

ISBN-10: 5747700995

ISBN-13: 9785747700994

This publication is written as a textbook for the process multidimensional geometryand linear algebra. At Mathematical division of Bashkir country college thiscourse is taught to the 1st yr scholars within the Spring semester. it's a half ofthe easy mathematical schooling. consequently, this direction is taught at actual andMathematical Departments in all Universities of Russia.

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**Example text**

This completes the proof of the theorem in whole. Let a ∈ ClU (b). This condition establishes some kind of dependence between two vectors a and b. This dependence is not strict: the condition a ∈ ClU (b) does not exclude the possibility that a′ ∈ ClU (b) for some other vector a′ . Such non-strict dependences in mathematics are described by the concept of binary relation (see details in [1] and [4]). Let’s write a ∼ b as an abbreviation for a ∈ ClU (b). 1 reveals the following properties of the binary relation a ∼ b, which is introduced just above: (1) reflexivity: a ∼ a; (2) symmetry: a ∼ b implies b ∼ a; (3) transitivity: a ∼ b and b ∼ c implies a ∼ c.

Let’s return to initial situation. Suppose that we have a mapping f : V → W that determines a matrix F upon choosing two bases e1 , . . , en and h1 , . . , hm in V and W respectively. The matrix F essentially depends on the choice of bases. In order to describe this dependence we consider four bases — two bases in V and other two bases in W . Suppose that S and P are direct transition matrices for that pairs of bases. Their components are defined as follows: m n ˜k = e j=1 ˜r = h Skj · ej , i=1 Pri · hi .

Instead of this we shall derive the result on partitioning V into the mutually non-intersecting cosets from the following theorem. 2. If two cosets ClU (a) and ClU (b) of a subspace U ⊂ V are intersecting, then they do coincide. Proof. Assume that the intersection of two cosets ClU (a) and ClU (b) is not empty. Then there is an element c belonging to both of them: c ∈ ClU (a) and c ∈ ClU (b). 1 we derive b ∈ ClU (c). 1, we get b ∈ ClU (a). 1. Let’s prove that two cosets ClU (a) and ClU (b) do coincide.

### Course of Linear Algebra and Multidimensional Geometry by Sharipov R.A.

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