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e2-solutions

# e2-solutions - Math 204 Exam 2 Name 1%— You have 50...

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Unformatted text preview: Math 204. Exam 2 October 20, 2010 Name: 1%— You have 50 minutes to complete the exam. Question Points 1 40 2 40 3 40 4 3O 5 50 6 bonus Total 1. Determine whether each of the following functions is a linear transformation. In the cases where it is linear, ﬁnd the matrix of the transformation. (a) T : R” —> R" given by T(§:’) = a? + 61 No " Than) —. H2): 2+\ =3§M E 9 T(\3‘\'TCI) ‘- C\.\-\) +(\+\) = L} 5° no} MW 2,“:— Ttoi = ”l. my 1w» WlWM‘J‘W M K; PM W TCo): o . (b) T : R" —> R" given by T(:E) = 353'— (52' - é'1)53‘. W wk.” n = 1) Tm ._ 3x _ 98‘ WUOK V's WW2 cowl Mt W Wm): TC'L): e—uaz Worm): 2T0): RC3—D=’+ +2.. 2. Which of the following linear transformations from R2 —) R2 is invertible. In the cases where it is invertibe, describe the inverse geometrically. Either ﬁnd the matrix of the inverse or give a formula for the inverse linear transformation. (a) A matrix obtained from I2 by an elementary row operation. —i 6:3 E=[0\] W05“ .2“ «aax l o 391.. E= :71»€"= {"371 > W1 v; x M’ S‘W‘ éémow 2E: ‘0 .. r M) m . w (b) A rotation through an angle of 0 radians. MVCNL in Q Ma“ W - 0' “Rab-3M4 AA= — “4-0) ‘\$"‘('°)] Locale) smW) \$\x[u9) Casi—9) "5\h(9) coslo) (c) A projection onto the line spanned by the unit vector 11'. huh-M No\ (“whw , Tod/u, Apnea-m '9’ MK " [0,0 3. Suppose that R9 is the linear transformation that rotates the plane R2 through 9 radians. Find all matrices that commute with R9. Be careful to distinguish the different cases that arise. 4. Give an example of a linear transformation T : R2 —> R2 that is not of the form ch for any number c and such that (a) (T5? - 53' = 0 for all choices of 50'. o T? :2». ELL] *1 ’ﬁ . *z :0 (b)T2=T. 2 .5? T: VWXFZ i’) Wc‘w. w M ‘3 +w WKJR . a. ”A,” :2 ' \'r0w\ QLJ.) A—‘2A 5. Prove the following: If B = {171,- . . ,ﬂ'm} is an orthonormal set, then Bis linearly independent. Tc SPruv-e- J- ——7 ,_., . l( C.\A\-\... +C‘M m“ =0 ) yew C‘=.--=CM=O v) '7 a) % SW09. C\\A,\ 4'..- + (“Auk ; 0 “M“ an \mAml— WM 3 C? TI . —-> -—-> —-> —-> \ |‘ \‘ + + CH KIM-q": :: 0-H" :O .2 .21. __7 ,3 - NM “NM =\) W‘Wx = l“ #6 6. (Bonus) How many linear transformations T : R2 -—) R2 are there with the following proper- ties. o The range of T is the line y = 2:. o T2 = T. Can you describe all such linear transformations? TM— [:3 Tia-(:3 T: Lab ab 7.. z az+oJo 03““ QB] T : T ‘57 ' q’ \9 Q14'Q\O aha-lo" bl 1% 6“ I W O‘Hasl 0":‘0 0.1+ QIL :. A \$7 Lag—:0; -;-‘> k" ‘IL (a *0 ...
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