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# bg224summer - "333d Boéazici University Department of...

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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 rank-nullity 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 MMjMAMgLMZ-HM, 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 Sdihw-TA-mrg Ml '—;5 T (TA) wO l5? maﬁa-plank 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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bg224summer - "333d Boéazici University Department of...

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