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## MODERN ALGEBRA

by: Estevan Champlin Sr.

44

0

3

# MODERN ALGEBRA MATH 220A

Estevan Champlin Sr.
UCSB
GPA 3.66

D. Morrison

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COURSE
PROF.
D. Morrison
TYPE
Class Notes
PAGES
3
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 3 page Class Notes was uploaded by Estevan Champlin Sr. on Thursday October 22, 2015. The Class Notes belongs to MATH 220A at University of California Santa Barbara taught by D. Morrison in Fall. Since its upload, it has received 44 views. For similar materials see /class/226879/math-220a-university-of-california-santa-barbara in Mathematics (M) at University of California Santa Barbara.

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Date Created: 10/22/15
MATH 220A NOVEMBER 28 2008 Example of a Smith Normal Form Computation This is the example worked in class on Monday7 November 24 We start with the matrix 132 68 68 A 78 76 40 78 112 40 which we wish to put into Smith normal form The rst step is to carry out the division algorithm on the pair 1327 787 keeping the results in matrix form First we subtract 78 from 132 1 1 132 54 0 1 78 T 78 39 Then we subtract 54 from 78 11112113 Next we subtract 2 gtlt 24 from 54 1313113111541 Finally7 we subtract 4 gtlt 6 from 24 1231151131 The combined row operation is 123113 3111311 511132111 25111351131 To use this to simply our matrix A7 we introduce the row operation matrix I 00 3 75 0 B1 713 22 0 0 0 1 Then 6 7176 4 BlA 0 788 74 78 112 40 1 2 The next row operation is easier subtract 13 times the rst row from the third which is accomplished by the matrix 1 0 0 B2 7 0 1 0 713 0 1 yielding 6 7176 4 BgBlA l 0 788 74 l 0 2400 712 J Now we need to apply the division algorithm to the pair 6 7176 whose steps we summarize more brie y lSOTllfmllil l fllillil ll llillil The combined row operation is 1 0 1 71 1 0 6 729 71 6 2 i2 1 0 1 30 1 7176 7 88 3 7176 7 0 39 To use this as a column operation in our Smith normal form algorithm we take the transpose and introduce 729 88 0 01 71 3 0 0 0 1 J so that 2 0 4 l BgBlAOl 7788 2364 i4 72400 7200 712 The next column operation is easier we want to subtract twice the rst column from the third column which is accomplished by the matrix 1072 02 010 l001l yielding 2 0 0 B2B1A0102l 7788 2364 1572 72400 7200 4788 Now we go back to row operations a simple row operation with matrix 1 0 0 B3 394 1 0 1200 0 1 yields 2 0 0 BngBlelog 0 2364 1572 0 7200 4788 We need to apply the division algorithm one last time7 to the pair 23647200 The steps this time are 1 0 2364 2364 1 721 2364 96 31720010801108108 1 0 96 96 178 96 0 0 1 0 12 11H1osll12 01H12ll12lg 10H12llol and the combined row operation is 101 1 l fl 13 Tl l 511 12 311550 which satis es 767 22 2364 12 7600 197 7200 7 0 39 The row operation matrix for the Smith normal form problem is then 1 0 0 B4 0 767 22 0 7600 197 which yields 2 0 0 B4B3B2B1A0102 0 12 12 0 0 36

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