Unformatted text preview: S into a vector space W extends uniquely to a linear map T W V ! W . Proof. We will assume that S D f v 1 ;:::;v n g is ﬁnite, in order to simplify the notation, although the result is equally valid if S is inﬁnite. Let f W S ! W be a function. Any element v 2 V has a unique expression as a linear combination of elements of S , v D P i ˛ i v i . There is only one possible way to deﬁne T.v/ , namely T.v/ D P i ˛ i f.v i / . It is then straightforward to check that T is linear. n Direct sums and complements The (external) direct sum of several vectors spaces V 1 , V 2 , . . . , V n over a ﬁeld K is the Cartesian product V 1 ² V 2 ²³³³² V n with component–by– component operations: .a 1 ;a 2 ;:::;a n / C .b 1 ;b 2 ;:::;b n / D .a 1 C b 1 ;a 2 C b 2 ;:::;a n C b n //...
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Vector Space

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