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Unformatted text preview: Doctoral Qualifying Examination Fall 1999 Advanced Calculus & Linear Algebra Name: 1D#: Start your answer on each question sheet. Attach all extra sheets you may have used to
the appropriate sheet. Hand in all question sheets. Time: 8:30 a 10:30 AM, September 1, 1999
Place: 8138 N106 ID #: 1‘ L01, f(3:,y} be a real—valued continuous function deﬁned in a. region RFurtheI‘. ﬁx, 1;)
has contirmous parLial derivatives of second order (including mixed dcrixratixres f“, and
fyz). Prove that fsvy : fyz: at. every interior poinL of the region. \’\"l'1a.t may happen at the boundary of the region
R? ‘ ' ID #: 2. Anal}'Lically (as opposed L0 graphically) determine the point on the ellipse
«41:31:2 + y? = 4, which is closest L0 the straight line :C + y = 10. ID #: 3. Let {5”} denote a sequence of real numbers which converges Lo 3 real number (1. Deﬁne
a, new sequence on as follows : ' 81+Sg‘l"'+8}¢_1+5k
k: 3 k:2,3, {T1 : 5'], ok :. Prove thaL the sequence {0k} also converges to o; ID #: 4. Consider the matrix mapping A : R4 —> R3 where l 3 6 1
A = 1 1 —1 —2
3 5 :1 —3 Find a basis and the dimension of (a) the image (if/1 and (b) the kernel 0M. 1D #: .5. Calculate the lengths of the major and the minor axes of the ellipse 17 3:2 +12 my +817}? :
4 . 1D it Let V be a ﬁnite—dimensional linear vector Space and let U be a linear vector space (not
necessarily of ﬁnite dimension). Let T : V ~:v U be a linear mapping. Prove that dimﬂ") ; diarnUi'er‘TJ + dimﬂm T), where Ker T and lmﬂ" are respectively the kernel and the image of T. ...
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This note was uploaded on 01/31/2011 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.
 Fall '08
 Feinberg,E

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