College Algebra Exam Review 364

College Algebra Exam Review 364 - a is strictly smaller...

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374 8. MODULES sequence of elementary column operations. Two matrices are equivalent if one is transformed into the other by a sequence of elementary row and column operations. We need a way to measure the size of a nonzero element of R . If R is a Euclidean domain, we can use the Euclidean function d . If R is non- Euclidean, we need another measure of size. Since R is a unique factoriza- tion domain, each nonzero element a can be factored as a D up 1 p 2 ±±± p ` . where u is a unit and the p i ’s are irreducibles. The number ` of irre- ducibles appearing in such a factorization is uniquely determined. We de- fine the length of a to be ` . For a a nonzero element of R , define j a j D ( d.a/ if R is Euclidean with Euclidean function d . length .a/ if R is not Euclidean. Lemma 8.4.7. (a) j ab j ² max fj a j ; j b jg . (b) j a j D j b j if a and b are associates. (c) If j a j ³ j b j and a does not divide b , then any greatest common divisor ı of a and b satisfies j ı j < j a j . Proof. Exercise 8.4.3 n In the following discussion, when we say that a is smaller than b , we mean that j a j ³ j b j ; when we say that
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Unformatted text preview: a is strictly smaller than b , we mean that j a j &lt; j b j . Lemma 8.4.8. Suppose that A has nonzero entry in the .1;1/ position. (a) If there is a element in the rst row or column that is not di-visible by , then A is equivalent to a matrix with smaller .1;1/ entry. (b) If divides all entries in the rst row and column, then A is equivalent to a matrix with .1;1/ entry equal to and all other entries in the rst row and column equal to zero. Proof. Suppose that A has an entry is in the rst column, in the .i;1/ position and that is not divisible by . Any greatest common of and satses j j &lt; j j , by the previous lemma. There exist s;t 2 R such that D s C t . Consider the matrix s t = = . This matrix has determinant equal to 1 , so it is invertible in Mat 2 .R/ . Notice that...
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