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Unformatted text preview: "333d Boéazici University, Department of Mathematics
Linear Algebra I, Summer 2008 I Date: July 16,2008 Midterm Test Name: HSAE M2 QZLEK I
Time Allowed: 120 min SID: ‘
Z = /105
"’ the? 1. Suppose {111,012, . . . , an} is a linearly independent set in the vector space V, and [3 E V. Prove 
that if {a1+ﬂ, a2+ﬂ, . . . , an+,6} is a linearly dependent set in V then ,8 E span {01, a2, . . . , an}. km}: Ceoﬂf well; way} C’ﬁvo We 'Old\+"+on°‘n"/'O 0’“: Ox: — ~ch 70 $M& {oh}: ~ . A,“ is 09%”?
“\th  KW? bankctxcls {13 M. (gin/J" lek ak least my, Cl’s Ts AM%% ' ’2: (cv'évc.)>o<\ 2r ~» + (C4 (00) 04“
wet areas N ﬁg 324% §o<~mm7m f w ' ‘V‘WY a Let AligC’éWl owl""ce‘ifim "(4+oea)u+m+cex,h . W9, rs Cd!” Astigﬂid ' 1
0 o  “has:
, €13 M, * 5334?". Math 224/ Mid — Page 2 of 4 — Name: 2. Let V be the vector space of all 2 x 2 matrices over the ﬁeld IF. Let also W1={AeV:A11+A12=0} and W2={AEV:A11+A21=0}3 be two subsets of V. ~
(a) (7 pts) Show that W1 and W2 are subspaces of V. Find their dimensions. A)" +63.” +A41+CE41
‘W/QM—r A42 + C(éafglz) :1] 'ﬂi‘1‘2]'[ H37] 15 a'basﬁ {0/ [NOV .—
v” =9 W4 ‘3 0 ) Lt {f 0 ’1 O O :1. (b) (8 pts) Find the dimensions of.W1 + W2 and W1 n W2. Is W1 + W2 = V?
A44 + A42 90 MA I444 New =0 "“ An?“ AQWANWZ '71! AZHWAM
5a M mm: m a w We you; i ‘1
mysusm::1tz$333}?me
m gleam/“WE e5 SW1: \mm’i a} VQWﬁl/y as \1 g. wwz ==> WWW; J 3. Let ]F be a field. Show that there does not exist a linear transformation T 2~]F5 —> F2 with
kerT = {($1,32,x3,m4,x5) E ]F : 11:1 = 32:2 and 1:3 = $4 = $5}. Hint: Apply the ranknullity theorem. 8% W X€WT é? Xm (3223‘7/L2’763;7¥3,Zz> :g: (3:*§;Qx’92?) ~+ “it; ((321314; 4» ’3) \‘Q, vmkvm’t‘t Wm diw‘mx‘iiw‘a
but MMjMAMgLMZHM, swume M MRL "ind , €23i‘vﬁ . .,
Math 224/Mid if”; 3 of 4 — Name: 4. Let V be a vector space and T e £(V, V). Prove that kerT n ImT = {0} if and only if 
T(Ta) = 0 implies Ta = 0 for each a e V. (=5) my?» WTﬂIm T 30$. (1% «C V :1. “r {rm} 5,0
:5 Tee;th owl ales/L3 might—r % TMQWTiAlM‘T ,2? 113,0 Uzi 5% axév SdihwTAmrg Ml '—;5 T (TA) wO l5? maﬁaplank Tet. $0 Cyan. $9.0 / 5. Let P2 be the vector space of all polynomials of degree at most 2 over 1R. Now deﬁne a linear 
mapT:’P2—"P2by T(:I:2) = a: + 1, T(:1:) = 2:2 — 1 and T(1) = 3. Let B = {x2,a:, 1} and B’ = {1,1 + z, 1 +51: +29} be two ordered bases for P2. First argue that
this T is Uniquely determined. Next, ﬁnd the representation matrices [T] B and [T] B, of T.
Hint: You should recall the isomorphism between ﬁnite dimensional vector spaces of the same dimension. nix“, ml) is at {Br mg M May 9(3ng “5 wmeéA} we 3mm {Mi g3 (MIT/1% Wmihtd‘ \INi 0413‘? WM 9 Tm :4» 2w
TLHQQ : x344; =: Tl4+><+><1l=$~ W + X14 *5 s >551» M 3 =1 9.1 + (*1) mx )4 “I. U“ M") 2.4 + 4. (Malawi lama, , _ r ’ ‘7” ~11 :
O {L 1 3m?
; 9.32% . 'haiﬁt'. — Page 4 of 4 — Name: Math 224/Mid 6. Let V and W be ﬁnite dimensional vector spaces over the ﬁeld 1F. Prove that there exists an  onto linear transformation T E £(V, W) if and only if dimV 2 dim W. (=—>) swam _ a T: V«rw may and We, Wt W AM on +th MT saw V
we MM WT #«éAMW évwa ﬁ'h}: Ond‘o .
50) MW WT 3O =9 ‘ WL “W” W" 654‘“ “T ‘ V "’ W
3% ~;j"(°(,)spt (for LSM M TM) 5 0 9w L>w
WC CM 3‘94! . a T 6% mg w.
W”! we, WW 834411», a T 8X8ch wk,
00¢ cm 5% W T '5 cheat? ' M4,; — The Exam Ends Here — ...
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This note was uploaded on 11/08/2011 for the course MATH 224 taught by Professor Gurel during the Spring '05 term at Boğaziçi University.
 Spring '05
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