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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 ) do we have up to isomorphism ? (10 points) 2. Let A be the following 3 × 3 matrix A = 0 2 1 - 4 6 2 4 - 4 0 . (i) Find the rational canonical form C of A , and an invertible matrix S such that C = S - 1 AS . (10 points) (ii) Find the Jordan canonical form J of A , and an invertible matrix T such that J = T - 1 AT . (10 points) 3. Let A be a 3 × 3 matrix whose Jordan canonical form is - 2 0 0 0 7 1 0 0 7 . Let B be a 4 × 4 matrix whose Jordan canonical form is 7 1 0 0 0 7 1 0 0 0 7 0 0 0 0 7 . 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 . (20 points) 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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