Algebra1993Aug - Written Ph D Qualifying Examination in...

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Unformatted text preview: Written Ph. D. Qualifying Examination in Linear Algebra. (MAf635) ' Fall 1993 Page I out of '2 I i. Let A be a real symmetric matrix satisfying Ak==1 for some positive 3 integer k, where l is the identity matrix of the same size as A. Prove A'ul. 3. Let 1' be a nonzero vector of the Euclidean space iRn. Let T: an~an be the linear operator given by the formula Ttnr)=r—2(I, vlv for all 163R”, where ( , ) is the standard inner product. Prove that ’I' can be represented by the matrix 10 0—1 ’ where 1 is the (n—1)><ln—-1) identity matrix. 3. Let T: lRJ—JR‘3 be the linear operator represented by the matrix with respect to the standard basis. Show there exist nonzero T-invariant subspaces U and l’ of R3 satisfying R3=ULBVX 4. Let T: Rn—«Rm be a linear transformation of rank k. Show there exist linear transformations U: kin-le and V: iRk—~!Rm, where U is onto and l-' is one-to- one, satisfying T=VU. Vt Fall 1993 D. Denote by Matnxllx‘) the set of all real Vn‘xn NEMatnflR; is called nilpotent if Nk 'ritten Ph. D. Qualifying Examination in Linear Algebra (MAT 63S) " Page .-. out of 2‘ matrices. A matrix =0 for some positive integer k. (a) Do all nilpotent matrices form a subspace of MatnilR)? (b) (c) Show I+N is diagonalizable if and only if N=O.. 6. Let A=[a2-j] be the nLvm. real matrix satisfying ai- J n. Denote by the same letter A the linear operator Rn—oan matrix with respect to the standard basis is A. (a) Describe Ker A and lm A as subsets of ER". (b) What is the minimal polynomial of :1? (c) Show A is diagonalizable. Prove 1+N is invertible, where IEMatnllR) is the identity matrix. =1 for all i, j==1, ..., whose representation ...
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This note was uploaded on 06/19/2011 for the course MATH 699 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.

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Algebra1993Aug - Written Ph D Qualifying Examination in...

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