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# FINAL_ANSWER_KEY - i 1 MATH 224 FINAL EXAM June 5 20.10...

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i MATH 224 FINAL- EXAM June 5, 20.10, 10:00-12:30 NAME and SURNAME: k.SY ( In ink) SIGNATUU: . 1 i /10 ii /12 III /14 LV /10 V /16 Vi /18 Total /80 ~ ... 1. Let V= Pi (x) be the real vector space of all polynomials of degree at most 2. Let T be a Iinear operatar on V defined by T(p(x) =.p(O)x+ p(1), Vp(x) E V. Find kernel and the range of T. (10 pts) So -rCfCxi1=:o ~ C,=-0 ik Ci +Cz. t-C~ -:O Ci. =--0 &. C3 =-c~ kare TJ -=- i P('X.) E- V ~ püd= c. (x-x 2.) } cElR. 3 dZCAK\ (. n:::. i o.x + b L a.,bE-R1

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2 2. Let V = C( [0,1 b be the veetor spaee of eontinuous real-valued funetions defined on the interval [0,1]' Let T be a linear operator on V defined by x (Tf)(x) = ff(t)dt for fE V. o Let ivi = U"E V: f is differentiable } and Wi = if E V: f(,Yi) = O subspaees of V. Decide ifthese subspaees are T -invariant or not. } be two. (12 pts) x A-. [CT-f-/(x}l = ~ ( -?(tJdt dx ~ dx.J O =~CX) ~ "XI, tc Wi er"~ ttco.t k+ f E- \1<12- k f(+) ~ 2t -1) OL+~t ~ fC"h.) =: O. Bltt- \ CT~) (~)==- SYZC2-t-IJdt =-~ =10 v So Tf f- W-z- 1hu.~ WL \<s. not T-\n""'rfan-t:
) 3 3.a) Let A be an nxn matrix of complex numbers so that Ois the only eigenvalue of A. Show that A is nilpotent.( Note: Your proofshould not exceed 2lines) (6 pts) /1

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FINAL_ANSWER_KEY - i 1 MATH 224 FINAL EXAM June 5 20.10...

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