College Algebra Exam Review 149

College Algebra Exam Review 149 - x;y 2 V and T.˛x/ D...

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3.3. VECTOR SPACES 159 Let’s make a few elementary deductions from the vector space axioms: Note that the distributive law ˛.v C w/ D ˛v C ˛w says that the map L ˛ W v 7! ˛v is a group homomorphism of .V; C / to itself. It follows that L ˛ .0/ D 0 and L ˛ . ± v/ D ± L ˛ .v/ for any v 2 V . This translates to ˛ 0 D 0 and ˛. ± v/ D ± .˛v/ . Similarly, C ˇ/v D ˛v C ˇv says that R v W ˛ 7! ˛v is a group ho- momorphism of .K; C / to .V; C / . Consequently, 0v D 0 , and . ± ˛/v D ± .˛v/ . In particular, . ± 1/v D ± .1v/ D ± v . Lemma 3.3.3. Let V be a vector space over the field K . then for all ˛ 2 K and v 2 V , (a) 0v D ˛0 D 0 . (b) ˛. ± v/ D ± .˛v/ D . ± ˛/v . (c) . ± 1/v D ± v . (d) If ˛ ¤ 0 and v ¤ 0 , then ˛v ¤ 0 . Proof. Parts (a) through (c) were proved above. For (d), suppose ˛ ¤ 0 but ˛v D 0 . Then 0 D ˛ ± 1 0 D ˛ ± 1 .˛v/ D ± 1 ˛/v D 1v D v: n Definition 3.3.4. Let V and W be vector spaces over K . A map T W V ! W is called a linear transformation or linear map if T.x C y/ D T.x/ C T.y/ for all
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Unformatted text preview: x;y 2 V and T.˛x/ D ˛T.x/ for all ˛ 2 K and x 2 V . An endomorphism of a vector space V is a linear transformation T W V ! V . The kernel of linear transformation T W V ! W is f v 2 V W T.v/ D g . The range of T is T.V / . Example 3.3.5. (a) Fix a polynomial f.x/ 2 KŒxŁ . The map g.x/ 7! f.x/g.x/ is a linear transformation from KŒxŁ into KŒxŁ . (b) The formal derivative P k ˛ k x k 7! P k k˛ k x k ± 1 is a linear transformation from KŒxŁ into KŒxŁ . (c) Let V denote the complex vector space of C –valued continuous functions on the interval Œ0;1Ł . The map f 7! f.1=2/ is a linear transformation from V to C ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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