MA554_JAN00

# MA554_JAN00 - F with | F |> 2 and A ∈ L V,V Show that...

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QUALIFYING EXAMINATION JANUARY 2000 MATH 554 - Prof. Wang Each problem is worth 10 points. 1. Let AX = B and A 1 X = B 1 be two consistent systems of linear equations. If they have the same set of solutions, prove that they are equivalent. 2. Let V be an n -dimensional subspace of Q [ X ]ov e r Q . Prove that there exist f 1 ,...,f n V and positive integers m 1 ,...,m n such that f i ( m j )= δ ij for 1 i,j n . 3. Let A be a linear operator on a ﬁnite dimensional vector space V over a ﬁeld F . Show that rank( A 2 )+rank( A 7 ) rank( A 5 )+rank( A 4 ) . 4. Let F be a ﬁeld and A,B M nn ( F ). Show that AB and BA have the same characteristic polynomial. 5. Let V be a ﬁnite dimensional vector space over a ﬁeld of characteristic 0, A L ( V,V )and T A the linear operator on L ( V,V )g ivenby T A ( B )= AB - BA . Assume that B is a characteristic vector of T A with nonzero characteristic value. Show that B is nilpotent.
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Unformatted text preview: F with | F | > 2 and A ∈ L ( V,V ). Show that there exist B,C ∈ L ( V,V ) such that (i) A = B + C , (ii) both B and C have cyclic vectors. 7. Let A ∈ M 6 × 6 ( Q ) satisfying A 3 = I . Write out the possible rational forms for A . 8. Let A ∈ M nn ( R ) satisfying A t A = AA t . Show that there exists a real polynomial f ( X ) such that f ( A ) = A t . 9. Let A,B ∈ M nn ( C ). Assume that A * = A , B * = B , tr( A ) = tr( B ) and X * AX ≥ X * BX for all X ∈ M n × 1 ( C ). Show that A = B . 10. Let F be a ﬁeld of characteristic 2. Give an example of a vector space V over F and distinct projections E 1 ,E 2 ,E 3 of V such that (i) E 1 + E 2 + E 3 = I (ii) E i E j 6 = 0 for i 6 = j ....
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## This document was uploaded on 01/25/2012.

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