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Math 260 Linear Algebra Midterm Exam II 05.12.2013 Last Name Dept. / Sec.: Signature
Name 3 Time I 17: 40
Student Ne Duration '70 minutes 5 QUESTIONS ON5 4 PAGES TOTAL 100 POINTS Q1 (15: 5—1—10 p.) ConsiderTE EURB) T(IE1,Z)= (ZyI—Z .T+z. a:+2y). a) Show that T IS an isomorphism. B
UL 2% Suttwcsto . Erma M mm m
§:{%%}%)EM1CT) Eff; 035%.}:0 . In “HHS (153%) ZSJX 4—Esﬂ b) Find the inverse transformation T‘l. Sky/EVE rt/LL W1“ Sjsteem OQgJ'E’Z—tg ’ 'M W x+—”E : E3
Kings : C
, ‘_ __’ Mr tom—614$, Bu"
We 22C. _
3 art9+1;— — adb+c K 1:: j 2:: E’X
b w bradC, ____ CUE—iv"<—~ Heli’lLL
F :3; _, “a“ #0 b"—
____:t E _._.ca; + L H; :5: {EL—E;
t (a, )C) I ”T J a; J 3. Q2 (25:15+10 p.) Considfar T: P3 (R) 9 R2, T(p (13)) = (3p(0) , f2}? (3:) (17:12). 0 a) Find the matrix III(._f }(T) of T relative to the pair of bases 6 — (1 255' 33:2 4.133) for
793R ): andf= ((1, ) ( 0))f0I1R2 MR 1191 TU):(3&1:&W§1+1 T<&x):
:10) q): Q‘ﬂ 951,1 (3)8921“); Zh’Eft 3f?“ TitHg) :{0)‘6)316f116£1_
Mm , 1.1 :2 £1? 3 '49
”(anujﬁ 1 14,5 Hg ~16 1)) Find a basis for the subspace ker (T) in 733 (R).
Take, Fm: a0 HZ i2 +62Z x 1.12319 C WCT) TL“ IP10} :0 and :Ffﬂdx: agiz + {XL}. 1:193 Q3 (2525+20 p.) Consider T : R3 ——> R2, T($,y?z) 2 (a: + 2,; — y). a) Find the matrix Mtg?” (T) with respect to the standard bases 6 and f fer R3 and R2, respectively. _ W”? O 1
“(€19CI)PLQ v! 11 b) Using the Change of Base Formula, ﬁnd the matrix Adler,” (T) with respect the pair of
bases 6’ =(e’11e’21e'3)a11d f’ 2 (fig), Where 6,120,010), 6’2 = “LL—1), e; = (0,2,w1),
and fi 1(1/21—1/2), fé= (1/11/53) We [Wt NEE/QC”: New”) Mﬁe,€)CT) “(éseﬂm \ ML: Jam, name): i; (f E] W 0 mi ~i Men? "‘ 1 4
' __ 1 ~E 1 O 1 Q 0 0
“(6&5 CT)”L1 1] [a W1 A 0 f «2
O wi “I
__ 4 ’i O i 0 0 A 2 1 02]
' 4 ~g 02 O 4 ‘3» ‘— 5’1 3 —‘«1 Q4 (25 p.) Let V = R4 and W = {9: + y + z — w = 0} be a 3—din1ensional subspace
in V. Consider the projection P E £(V) onto the subspace W parallei to the vector
v: (1 0 G. 0) Find the matrix MQE a) (P) of P reiative to the standard basis 6 for V Choose 0 EQSES fw‘Ezfé #01" 47k sub—rinse W  1 o o a BM,
MHA?)CP)— 0 i o (:2 T
0 O i C}
0 O 0 0 Q5 (10 p.) Let V be a vector space (of any dimension), and T e S (V) a nilpotent iinear
transformation such that T’“ = 0 and CW” # 0 for k > 1. Show that ker (T) 75 {0} and HIT(JeéV Vév' TL?” TCTMU’H) T (0:) OytLatLi:
T"“( )é— mm)
M MCT): V 4:,me Vgev jxem 3*T(><) f—k—g =—> 7““(3) : 770:0 2 TM *2 i 3 0/ ...
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 Linear Algebra, basis, algebra midterm exam, Student Ne Duration, Sky/EVE rt/LL W1, inverse transformation T‘l

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