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MA554_JAN06

# MA554_JAN06 - R is a ﬁeld and n ≤ m show that the...

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QUALIFYING EXAMINATION JANUARY 2006 MATH 554 - Prof. J. Wang 1. Let E i , i = 1 , · · · , ‘ be projections of a finite dimensional vector space V and α 1 , · · · , α scalars. If E i have the same range, then A = X i =1 α i E i satisfies A 2 = X i =1 α i ! A . 2. Let F be a field and S = { E M nn ( F ) | E 2 = E } . Show that the span of S is M nn ( F ). 3. Let A = [ A 1 , A 2 , A 3 , A 4 , A 5 ] be a 4 × 5 matrix. Assume that the general solution for AX = 0 is given by X = s 2 s - t t t + s 2 t . (a) Find a maximal independent subset B of { A 1 , A 2 , A 3 , A 4 , A 5 ). (b) Express A 1 + A 2 + A 3 + A 4 + A 5 as a linear combination of B . 4. Let K be a field and D : M nn ( K ) K be a function such that D ( AB ) = D ( A ) D ( B ) and D ( I ) 6 = D (0). Show that if rank( A ) < n , then D ( A ) = 0. 5. Let R be a commutative ring with identity, A M mn ( R ), B M nm ( R ) and I the identity matrix. (a) | I m - AB | = | I n

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Unformatted text preview: R is a ﬁeld and n ≤ m , show that the characteristic polynomials p AB ( x ) and p BA ( x ) of AB and BA respectively satisfy p AB ( x ) = x m-n p BA ( x ). 6. Let A = [ a ij ] be the ( n + 1) × ( n + 1) matrix with a ij = ( i + j-2)! and 0! = 1. (Hint: A = LDL T ) (a) A is positive deﬁnite. (b) det A = (0!1! ··· n !) 2 . (c) ( n !) 2 A-1 ∈ M ( n +1)( n +1) ( Z ). 1 7. Let A be an n × n matrix over C and p be a prime number. Suppose that I 6 = A and A p = I and tr( A ) = positive integer ‘ . Show that n = ‘ + sp with s a positive integer. 2...
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MA554_JAN06 - R is a ﬁeld and n ≤ m show that the...

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