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

By Sharipov R.A.

This booklet 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. accordingly, this direction is taught at actual andMathematical Departments in all Universities of Russia.

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There is a similar proposition for Ker f . 3. A linear mapping f : V → W is injective if and only if its kernel is zero, i. e. Ker f = {0}. Proof. Let f be injective and let v ∈ Ker f . Then f (0) = 0 and f (v) = 0. But if v = 0, then due to injectivity of f it would be f (v) = f (0). Hence, v = 0. This means that the kernel of f consists of the only one element: Ker f = {0}. Now conversely, suppose that Ker f = {0}. Let’s consider two different vectors v1 = v2 in V . Then v1 − v2 = 0 and v1 − v2 ∈ Ker f .

S · es + γ1 · es+1 + . . + γn−s · en = 0. This is the linear combination of basis vectors of V , which is equal to zero. Basis vectors e1 , . . , en are linearly independent. Hence, this linear combination is trivial and γ1 = . . = γn−s = 0. 5). 4). The theorem is proved. § 8. Linear mappings. 1. Let V and W be two linear vector spaces over a numeric field K. A mapping f : V → W from the space V to the space W is called a linear mapping if the following two conditions are fulfilled: (1) f (v1 + v2 ) = f (v1 ) + f (v2 ) for any two vectors v1 , v2 ∈ V ; (2) f (α · v) = α · f (v) for any vector v ∈ V and for any number α ∈ K.

S · es + γ1 · es+1 + . . + γn−s · en = 0. This is the linear combination of basis vectors of V , which is equal to zero. Basis vectors e1 , . . , en are linearly independent. Hence, this linear combination is trivial and γ1 = . . = γn−s = 0. 5). 4). The theorem is proved. § 8. Linear mappings. 1. Let V and W be two linear vector spaces over a numeric field K. A mapping f : V → W from the space V to the space W is called a linear mapping if the following two conditions are fulfilled: (1) f (v1 + v2 ) = f (v1 ) + f (v2 ) for any two vectors v1 , v2 ∈ V ; (2) f (α · v) = α · f (v) for any vector v ∈ V and for any number α ∈ K.

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