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Fall 2009 Midterm #2 Solution

# Fall 2009 Midterm #2 Solution - Linear Algebra F09 Name...

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Unformatted text preview: Linear Algebra F09 Name: Page 1 of 7 (1) (5 marks) Find parametric equations for the line passing the point P (1, —1, 2) and parallel to the two planes given below: 233—y+3a:=0, \$—3y+3z—2=0 n‘. (1‘—\)3) ,0,=(4 “355) M,- “gamut-00.1w: a. ewe» Vmuts. want misc—vim: vectot A an. out. due . 1“: 015° 15mm. é=\ A = (6,-3,’5). l "3 3 so Vanna-:ch EQUATio us At; x: J 4.5-}: 3: -& -3‘k 2= 1 4k. (2) (4 marks) Find the steady state vector q for the following regular tran- sition matrix: P _ 1/4 1/5 _ 3/4 4/5 (JAN 1 so -1er , Lg. inl=Q “as. C1-P)1=Q_. i~ " 1—9 i W w .3/\‘ V: 256065 0 o 0 So UET 11:3,‘33 1\=%\$- _ is _. HAS] 5: \ Is- So - — s 2< Tamas i K ‘ 1?? 4| .—_ “11 (31' Stacy SM“? MG- ﬂ Lg/ l‘I 3| Linear Algebra F09 Name: Page 2 of 7 (3) (4 marks) Say T : R2 —> R2 is the linear operator deﬁned by: a reﬂection about the line y = —33 followed by a counter-clockwise rotation by an angle of 7T / 2 followed by an othogonal projection onto the y-axis. Use [T] = [T(e1) | T(e2)] to get the standard matrix of T. (NOTE: for full marks, you must use this theorem.) 3%? (4) (3 marks) Let T : R3 —> R3 be the linear operator deﬁned by T(a:1, 3:2, :63) = (2331 + 332 + 2333, —2331 + 2332 + 5333, 4331 + 2:132 + 7333) Determine if T is one-to—one. Linear Algebra F09 Name: Page 3 of 7 (5) (5 marks) Determine Whether the set S = {M1, M2, M3} is a basis for the vector space V of 2 X 2 symmetric matrices, Where 1 2 0—1 1 0 Mlzi 2 Oi’M2=[—1 oi’MFio 1i .3 Ts A 3Ang in? WCS)=— \i & st do. in. ® (,uuu. saw (53’ \/ LET A: 4 '5 BE 1% must 10 suck.) 1*?“ ARI \LU‘LM‘L3 so THAT b c A: LMd Lbei‘ql’t; “if. ‘L\ +\¢3=Q. Zl‘. = b lkf‘k}, . 5 k3 = C ,4 Wm wnu auancmo wmlu | o \ a. ﬂow @o a QrC _ 7_ —| o lo _,9 a Q -Zq,-Ho go 1.3% system 5 A-UUA'15 ;_ —\ o b Queue: 0 o (a) 5 Comment 0 O I C O o o 0 so SOADCS)‘ . . - l O I w G o a (,a-r system wmx 0:2?wa “mu” A 5 z -\ a a ed) 2 7. -—x c Luau: 9 ‘30 ° ° ‘ o o e :00 Fun: walnuts , So L=\¢,=L,=o is 1-“; out, «was»: , ~ so 3 \s uh. tun. So S \s A Bnéls Fog, Linear Algebra F09 Name: Page 4 of 7 (6) (6 marks) Let V be Euclidean 5-space, R5, and let W be the subset of all 5-tuples (a, b, c, d, 6) with the property that c = a — b and d = b — a. (i) Show that W is a subspace of V. G) crich (meow mum, Awf-n'ou : '9? 9);! , SW any {QUJ- = («may quﬁ gTqu'l) g ,\_1 m \U news (5 8" \l = C41,|01,43‘latgl91'q13€1) u-r V : (04+ «1 ’ ‘O\-§L)‘.,(au+41)- ’ "€\+el) so Lian; is {0 \U- ® (Heck. upsuw upoa. SLALM. Mum‘xl ~. ix: L1. Add) it is i6 [L ,SW M5”) u)- w {w U. AS AwE, .— ku__( k4,, , kb‘ , Itch—Um, Lbrhqukeo 50 L3 is in (ii) Give a basis of W and state the dimension of W stw . A wit: 09 UL} "v: {(uo,\,-\,o),(o,t.-\.\.o),C°.°.°,O.I35 Linear Algebra F09 Name: Page 5 of 7 (7) (8 marks) 2 6 1 1 7 1 3 0 0 2 1 3 3 0 —1 0 0 1 0 —1 LetA— 4 12 7 1 5 andRzrreﬂA) = 0 0 0 1 4 1 3 —1 —1 —1 0 0 0 0 0 (a) Give a basis for the rowspace of AT SK"? (b) Give a basis for the nullspace of A (C) State the following: (i) rank(A) 2 (ii) nullity(A) = (111) Tank(AT) = (iv) nullity(AT) = Linear Algebra F09 Name: Page 6 of 7 (8) (5 marks) True / False. Indicate Whether the following statements are always True or sometimes False. (a) Elementary row operations do not change the nullspace of a matrix 5K“? True False (b) If AX = 0 has only the trivial solution and A is n X n, then TCLTLk<A> = 71 Saw True False (c) If T(0) = 0, then T is a linear transformation Sui? True False (d) The planes 2a: — y + 3,2 — 2 = 0 and 3a: — 22 + 1 = 0 are perpendicular to each other <1,—\,3)-(‘5,0.-1)= é—eao Z/True False (e) With u,V E R”, then ||u+v||2 2 Mn —V||2 ifu J. V liz E/True False (as =0 (f) If A is an n X m matrix, and nullity(AT) = 2, then TCLTLk<A> = m — 2 SK“) True False (g) The set of all 4-tuples (a, b, c, d) with a = 1 is a subspace of R4 True YFalse (h) With 81,82 subsets of a vector space V, if span(81) = span(82), then 81 = 82 True E/False (i) If W is the space of n X n upper triangular matrices, then dim(W) = 712/ 2 \$69 True False (j) If {V1, V2,V3} is a linearly dependent set, then so is {V1,V2} True 3/ False ...
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Fall 2009 Midterm #2 Solution - Linear Algebra F09 Name...

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