By Sharipov R.A.

This booklet is written as a textbook for the process multidimensional geometryand linear algebra. At Mathematical division of Bashkir kingdom collage thiscourse is taught to the 1st 12 months scholars within the Spring semester. it's a half ofthe uncomplicated mathematical schooling. hence, this direction is taught at actual andMathematical Departments in all Universities of Russia.

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**Example text**

Xn → x1 · w1 + . . + xn · wn . Now it is easy to verify that the required mapping is the composition f = ϕ ◦ ψ. 42 CHAPTER I. LINEAR VECTOR SPACES AND LINEAR MAPPINGS. 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 .

Expanding x and y in some basis e1 , . . , en , we get the following formula relating their coordinates: y1 .. y n = F11 .. F1n . . Fn1 x1 .. · .. .. . xn . . 5) of Chapter I. From this formula we derive that x belong to the kernel of the operator f if and only if its coordinates x1 , . . , xn satisfy the homogeneous system of linear equations F11 .. F1n . . Fn1 x1 . .. · ... xn . . Fnn = 0 .. , . 7) 0 The matrix of this system of equations coincides with the matrix of the operator f in the basis e1 , .

So, the choice of bases in V and W defines an isomorphism of Hom(V, W ) and Km×n . , 2004. CHAPTER II LINEAR OPERATORS. § 1. Linear operators. The algebra of endomorphisms End(V ) and the group of automorphisms Aut(V ). A linear mapping f : V → V acting from a linear vector space V to the same vector space V is called a linear operator 1 . Linear operators are special forms of linear mappings. Therefore, we can apply to them all results of previous chapter. However, the less generality the more specific features.