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 nbyn 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 5dimensional 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 nbyn matrix A over a field Fis similar to a diagonal
matrix if and only if there is a basis for F‘"’, the space of nbyI
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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 Spring '11
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 Algebra

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