**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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