By Cohn H. M.

Stories very important history in algebra and introduces extra complicated subject matters, emphasizing linear algebra and the houses of teams and earrings. contains extra labored difficulties and an entire set of solutions to the routines. additionally gains accelerated proofs and extra in-depth remedies of affine areas, linear programming, duality, Jordan basic shape and workforce thought.

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**Example text**

Let a and b be positive integers. A common multiple of a and b is an integer N such that a and b both divide N . The least common multiple of a and b is the smallest positive common multiple of a and b. It is denoted by LCM(a, b). 26. Prove that a and b have a least common multiple. 27. If a has a prime factorization given by a = p1k1 · · · prkr , prove that any positive multiple M of a has a prime factorization given by a = p1m 1 · · · prm r q1n 1 · · · qsn s , where q1 , . . , qs are primes not in the list p1 , .

PROOF. Conclusion (a) is immediate from the deﬁnition. For (b), let σ = τ1 · · · τk with each τ j equal to a transposition. 23 recursively, using (a) at the end: sgn(τ1 · · · τk ) = (−1) sgn(τ1 · · · τk−1 ) = (−1)2 sgn(τ1 · · · τk−2 ) = · · · = (−1)k−1 sgn τ1 = (−1)k sgn 1 = (−1)k . 22 shows that any permutation is the product of transpositions. If σ is the product of k transpositions and τ is the product of l transpositions, then σ τ is manifestly the product of k + l transpositions. Thus (c) follows from (b).

Many readers will already be familiar with some aspects of this theory, particularly in the case of the vector spaces Qn , Rn , and Cn of column vectors, where the tools developed from row reduction allow one to introduce geometric notions and to view geometrically the set of solutions to a set of linear equations. Thus we shall 33 II. Vector Spaces over Q, R, and C 34 be brief about many of these matters, concentrating on the algebraic aspects of the theory. Let F denote any of Q, R, or C. 1 A vector space over F is a set V with two operations, addition carrying V × V into V and scalar multiplication carrying F × V into V , with the following properties: (i) the operation of addition, written +, satisﬁes (a) v1 + (v2 + v3 ) = (v1 + v2 ) + v3 for all v1 , v2 , v3 in V (associative law), (b) there exists an element 0 in V with v + 0 = 0 + v = v for all v in V , (c) to each v in V corresponds an element −v in V such that v + (−v) = (−v) + v = 0, (d) v1 + v2 = v2 + v1 for all v1 and v2 in V (commutative law); (ii) the operation of scalar multiplication, written without a sign, satisﬁes (a) a(bv) = (ab)v for all v in V and all scalars a and b, (a) 1v = v for all v in V and for the scalar 1; (iii) the two operations are related by the distributive laws (a) a(v1 + v2 ) = av1 + av2 for all v1 and v2 in V and for all scalars a, (b) (a + b)v = av + bv for all v in V and all scalars a and b.