March 8, 2017

# An Introduction To Linear Algebra by Kuttler By Kuttler

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Algebra for College Students , Eighth Edition

Make math a snap with ALGEBRA for students. utilizing daily language and plenty of examples, Kaufman and Schwitters allow you to follow algebra techniques and ace the try. This quantity additionally comes with Interactive Skillbuilder CD-ROM. This software is full of over eight hours of video guide to aid all of it make feel.

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Are linearly independent if no linear combination of distinct Vu V2,... with nonzero coefficients is equal to the zero vector. Thus, the set {0} which is a nonempty subset of every vector space is linearly dependent. On the other hand, every one-element subset of a vector space which is not equal to {0} is linearly independent. Also, we have REMARK 1. If y is a subset of a vector space such that 0 e 9 then 9 is a linearly dependent set of vectors. In view of Definition 2 a necessary and sufficient condition for a non­ empty subset 9 of a vector space to be linearly independent is that 2 ^ =0 ifandonlyifj f = 0 (11) with Vie 9 and V{ 4= V3 for i 4= j .

Un be subspaces sional vector space V. Then of a finite dimen- r = %+%+> - + °un if and only ifT = °UX + °U2 + (- °lln and dim T = dim °UX H- dim ^ 2 + h dim ^rn. Based on the notion of isomorphism introduced on page 46 we prove the following theorem. T H E O R E M 8. Let °U and T be finite dimensional vector spaces over the same field. , W = T if and only if dim^ = dimr Proof Let °U = T and

1 and let {Uu U2> .

Prove that there are infinitely many elements qofQ such that q2 + 1 = 0. 15. Let SF be a field. Prove that a nonzero ideal of ^\X\ is a prime ideal of \$F\X\ if and only if it is generated by a monic irreducible element of 3F\X\. Prove also that every nontrivial prime ideal of 2F\X\ is a maximal ideal of£F\X\. 16. Prove that in a finite field with n elements every element is a root of the polynomial Xn — X. 17. Let 3F be a finite field with n elements and p be an irreducible element of degree m o f f [X].