Get Abstract Algebra (3rd Edition) PDF

By David S. Dummit, Richard M. Foote

ISBN-10: 0471433349

ISBN-13: 9780471433347

"Widely acclaimed algebra textual content. This e-book is designed to provide the reader perception into the facility and sweetness that accrues from a wealthy interaction among varied parts of arithmetic. The e-book rigorously develops the speculation of other algebraic constructions, starting from uncomplicated definitions to a few in-depth effects, utilizing a variety of examples and routines to help the reader's knowing. during this manner, readers achieve an appreciation for a way mathematical buildings and their interaction result in strong effects and insights in a few diversified settings."

Covers basically all undergraduate algebra. Searchable DJVU.

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Extra resources for Abstract Algebra (3rd Edition)

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Ur if each Ui is a subspace of V, and every element v of V has a unique expression of the form v ˆ u1 ‡ . . ‡ ur (ui P Ui ). Also, V ˆ U È W if and only if V ˆ U ‡ W and U ’ W ˆ {0}. 3. A linear transformation W: V 3 W satis®es (u ‡ v)W ˆ uW ‡ vW and (ëv)W ˆ ë(vW) for all u, v in V and all ë in F. Ker W is a subspace of V and Im W is a subspace of W, and dim V ˆ dim (Ker W) ‡ dim (Im W)X 4. A linear transformation W: V 3 W is invertible if and only if Ker W ˆ {0} and Im W ˆ W. 5. Given a basis B of the n-dimensional vector space V, there is a correspondence between the endomorphisms W of V and the n 3 n matrices [W]B over F.

We use the letters F and G in the name `FG-module' to indicate that V is a vector space over F and that G is the group from which we are taking the elements g to form the products v g (v P V). Note that conditions (1), (4) and (5) in the de®nition ensure that for all g P G, the function v 3 vg is an endomorphism of V. 3 De®nition Let V be an FG-module, and let B be a basis of V. For each g P G, let [ g]B denote the matrix of the endomorphism v 3 v g of V, relative to the basis B . The connection between FG-modules and representations of G over F is revealed in the following basic result.

12 Theorem Suppose that V is an FG-module with basis B , and let r be the representation of G over F de®ned by r: g 3 [ g]B ( g P G)X (1) If B 9 is a basis of V, then the representation ö: g 3 [ g]B 9 ( g P G) of G is equivalent to r. 23). 24), for all g P G, we have [ g]B ˆ T À1 [ g]B 9 TX Therefore ö is equivalent to r. (2) Suppose that r and ó are equivalent representations of G. Then for some invertible matrix T, we have gr ˆ T À1 ( gó )T for all g P GX Let B 0 be the basis of V such that the change of basis matrix from B to B 0 is T.

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Abstract Algebra (3rd Edition) by David S. Dummit, Richard M. Foote


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