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Algebra2004Jan - 1 2 Ph.D PRELIMINARY EXAMINATION SPRING...

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Unformatted text preview: 1 . 2. Ph.D. PRELIMINARY EXAMINATION - SPRING 2004 ALGEBRA Let A be a complex matrix with characteristic polynomial (x—l)“(x+2)". Assume that the rank of (xi—17)2 is 5 and the rank of (A+217) is 4. What is the Jordan canonical form of A? Justify your answer. (a) Let A be an n-by-n matrix over a field F. Then C(A)={XeMn(F)IXA=AX} is called WWMW). Let YeMn(F) be an invertible matrix. Show that C(YAY")=Y[C(A)]Y". (Note: Y[C(A)]Y“ = {YXY“ I X e C(A)}.) (b) If Fis the field complex numbers and n = 2, what is the smallest 3. (a) (b) 4. 5. 6. dimension C(A) can have? For each of the following statements, either prove it or give an example to show that it is false. Assume ¢:V—>W is a linear transformation between vector spaces. If {vl,v2 ...... ,vn} is a subset for V with {¢(v,),¢(v2), ..... ,¢(vn)} linearly independent in W, then {v1,v2, ..... ,vn} is linearly independent in V. Assume V is a 5-dimensional vector space and W is a 3 dimension vector space with XzV—aW and Y:V—>W surjective linear transformations. Then there exists veV, nonzero, such that X(v)=Y(v)=0. Show that an n-by-n matrix A over a field Fis similar to a diagonal matrix if and only if there is a basis for F‘"’, the space of n-by-I matrices over F, consisting of eigenvectors for A. Let P2 be the vector space of polynomials of degree at most 2 over the real numbers together with the inner product <flg>=J'o'fgdx. Let ¢:Pz—>R be the functional given by ¢(f)=f(1). Find gepz’ such that ¢(f)=<flg>, for all gePz. Let V be a finite dimensional inner product space over the complex numbers and let W be a subspace with orthonormal basis {051,052 ....... 05,}. If fieV, show that y=2<filai)ai is the unique element of W with “B " Y" = W minimal. ...
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