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

By Sharipov R.A.

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

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The linearity of the mapping Rh is interpreted as the linearity of this binary operation in its first argument, while the linearity of Lh is said to be the linearity of composition in its second argument. A binary algebraic operation linear in both arguments is called a bilinear operation. A situation, where a linear vector space is equipped with an additional bilinear algebraic operation, is rather typical. 1. A linear vector space A over a numeric field K equipped with a bilinear binary operation of vector multiplication is called an algebra over the field K or simply a K-algebra.

Now let’s consider the rest axioms (3) and (4). The zero mapping is the best pretender for the role of zero element in the space Map(V, W ), it maps each vector v ∈ V to zero vector of the space W . For this mapping we have (f + 0)(v) = f (v) + 0(v) = f (v) + 0 = f (v). As we see, the axiom (3) in Map(V, W ) is fulfilled. Suppose that f ∈ Map(V, W ). We define the opposite mapping f ′ for f as follows: f ′ = (−1) · f . Then we have (f + f ′ )(v) = (f + (−1) · f )(v) = f (v) + + ((−1) · f )(v) = f (v) + (−1) · f (v) = 0 = 0(v).

8 and consider the total set of vectors in them: e1 , . . , es , es+1 , . . , es+p , es+p+1 , . . , es+p+q . 10) form a basis in the sum of subspaces U1 + U2 . Let w be some arbitrary vector in U1 + U2 . 2) for this vector is written as w = u1 + u2 . Let’s expand the vectors u1 and u2 in the above two bases of the subspaces U1 and U2 respectively: p s u1 = i=1 αi · ei + s u2 = i=1 α ˜ i · ei + j=1 q j=1 βs+j · es+j , γs+p+j · es+p+j . 10). 10) is a spanning system of vectors in U1 + U2 . 10) is a linearly independent system of vectors we consider a linear combination of these vectors being equal to zero: q s+p i=1 αi · ei + i=1 αs+p+i · es+p+i = 0.

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