By Leonard E Dickson
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Extra info for Algebraic invariants
ALGEBRAIC OPERATIONS WITH MAPPINGS. 3. The pointwise product of a linear mapping f : V → W by a number α ∈ K is a linear mapping from the space V to the space W . Proof. Let h = f + g be the sum of two linear mappings f and g. following calculations prove the linearity of the mapping h: The h(v1 + v2 ) = f (v1 + v2) + g(v1 + v2 ) = (f(v1 )+ +f(v2 )) + ((g(v1 ) + g(v2 ) = (f(v1 )+ + g(v1 )) + (f(v2 ) + g(v2 )) = h(v1) + h(v2 ), h(β · v) = f(β · v) + g(β · v) = β · f(v)+ + β · g(v) = β · (f(v) + g(v) = β · h(v).
12) and prove that they form a basis in V . 6. 12). In order to prove the linear independence of these vectors we consider a linear combination of them being equal to zero: α1 · e1 + . . + αs · es + αs+1 · es+1 + . . + αs+r · es+r = 0. 13) and take into account that f(ei ) = hi for i = 1, . . , s. Other vectors belong to the kernel of the mapping f, therefore, f(es+i ) = 0 for i = 1, . . , r. 13) we derive α1 · h1 + . . + αs · hs = 0. The vectors h1 , . . , hs form a basis in Im f. They are linearly independent.
1 for the operation of coset addition. ˜ within the coset Q. Now let’s consider two different representatives v and v ˜ − v ∈ U . Hence, α · v ˜ − α · v = α · (˜ Then v v − v) ∈ U . 2 for the operation of multiplication of cosets by numbers. 5. 3) is a linear vector space. This space is called the factorspace or the quotient space of the space V over its subspace U . Proof. The proof of this theorem consists in verifying the axioms (1)-(8) of a linear vector space for V /U . The commutativity and associativity axioms for the operation of coset addition follow from the following calculations: ClU (v1 ) + ClU (v2 ) = ClU (v1 + v2 ) = = ClU (v2 + v1 ) = ClU (v2 ) + ClU (v1 ), (ClU (v1 ) + ClU (v2 )) + ClU (v3 ) = ClU (v1 + v2 ) + ClU (v3 ) = = ClU ((v1 + v2 ) + v3) = ClU (v1 + (v2 + v3 )) = ClU (v1 ) + ClU (v2 + v3 ) = ClU (v1 ) + (ClU (v2 ) + ClU (v3 )).