College Algebra Exam Review 148

# College Algebra Exam Review 148 - to the notion of an...

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158 3. PRODUCTS OF GROUPS in 1 g . S n acts on the group of diagonal matrices by permutation of the diagonal entries. One ﬁnal example shows that direct products and semidirect products do not exhaust the ways in which a normal subgroup N and the quotient group G=N can be ﬁt together to form a group G : 3.2.6. Z 4 has a subgroup isomorphic to Z 2 , namely the subgroup gener- ated by Œ2Ł . The quotient Z 4 =Z 2 is also isomorphic to Z 2 . Nevertheless, Z 4 is not a direct or semidirect product of two copies of Z 2 . 3.3. Vector Spaces You can use your experience with group theory to gain a new appreciation of linear algebra. In this section K denotes one of the ﬁelds Q , R ; C , or Z p , or any other favorite ﬁeld of yours. Deﬁnition 3.3.1. A vector space V over a ﬁeld K is a abelian group with a product K ± V ! V , .˛;v/ 7! ˛v satisfying the following conditions: (a) 1v D v for all v 2 V . (b) .˛ˇ/v D ˛.ˇv/ for all ˛;ˇ 2 K , v 2 V . (c) ˛.v C w/ D ˛v C ˛w for all ˛ 2 K and v;w 2 V . (d) C ˇ/v D ˛v C ˇv for all ˛;ˇ 2 K and v 2 V . Compare this deﬁnition with that contained in your linear algebra text; notice that we were able to state the deﬁnition more concisely by referring
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Unformatted text preview: to the notion of an abelian group. A vector space over K is also called a K –vector space. A vector space over R is also called a real vector space and a vector space over C a com-plex vector space. Example 3.3.2. (a) K n is a vector space over K , and any vector subspace of K n is a vector space over K . (b) The set of K –valued functions on a set X is a vector space over K , with pointwise addition of functions and the usual multipli-cation of functions by scalars. (c) The set of continuous real–valued functions on Œ0;1Ł (or, in fact, on any other metric or topological space) is a vector space over R with pointwise addition of functions and the usual multiplication of functions by scalars. (d) The set of polynomials KŒxŁ is a vector space over K , as is the set of polynomials of degree ² n , for any natural number n ....
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