This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: METU Northern Cyprus Campus
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/ ...
View
Full Document
 Winter '10
 uguz
 Linear Algebra, Algebra, basis, algebra midterm exam, Student Ne Duration, Sky/EVE rt/LL W1, inverse transformation T‘l

Click to edit the document details