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Unformatted text preview: X i T i denes a homomorphism from Kx to End K .V / . What does this mean, and how does it follow from Proposition 6.2.5 ? End K .V / is a vector space over K as well as a ring. The product of a scalar 2 K and a linear map S 2 End K .V / is dened by .S/.v/ D S.v/ for v 2 V . Let I denote the identity endomorphism of V dened by I.v/ D v . Then I.v/ D v for v 2 V . The map ' W K ! End K .V / given by 7! I is easily seen to be a unital ring homomorphism from K to End K .V / . By Proposition 6.2.5 , there is a unique homomorphism ' T W Kx ! End K .V / with ' T .x/ D T and ' T ./ D I . Moreover, ' T . P i i x i / D P i . i I/T i D P i i T i . We usually write p.T / for ' T .p/ ....
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 Fall '08
 EVERAGE
 Algebra, Polynomials

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