MA554_JAN02

MA554_JAN02 - QUALIFYING EXAMINATION JANUARY 2002 MATH 554...

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QUALIFYING EXAMINATION JANUARY 2002 MATH 554 - PROF. MATSUKI Name ID # 1. Let A be an Abelian group satisfying conditions ( α )(5 3 · 7 4 ) A =0,and ( β )# { a A ;(5 · 7) a =0 } =5 2 · 7 2 . (i) Show that A is finitely generated. (10 points) (ii) How many such Abelian groups that satisfy conditions ( α )and( β )dowe have up to isomorphism ? (10 points) 2. Let A be the following 3 × 3 matrix A = 02 1 - 462 4 - 40 . (i) Find the rational canonical form C of A , and an invertible matrix S such that C = S - 1 AS .( 1 0 p o i n t s ) (ii) Find the Jordan canonical form J of A , and an invertible matrix T such that J = T - 1 AT 1 0 p o i n t s ) 3. Let A be a 3 × 3 matrix whose Jordan canonical form is - 200 071 007 . Let B be a 4 × 4 matrix whose Jordan canonical form is 7100 0710 0070 0007 . Consider the space V of 3 × 4 matrices X with entries in C such that AX = XB. Find the dimension of the vector space V over C 2 0 p o i n t s ) Typeset by A M S -T E X 1
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2 4. Let V = C 4 be a 4-dimensional vector space over C and let f : V V be a
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MA554_JAN02 - QUALIFYING EXAMINATION JANUARY 2002 MATH 554...

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