Fall 2010 Midterm #2 Solutions (Mihai's class)

# Fall 2010 Midterm #2 Solutions (Mihai's class) -...

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Unformatted text preview: MATH1850/2050 s09 Name: SOLUTKM Page 1 of 7 (1) (8 marks) Answer the following questions in the space provided: (a) Write an equation of the plane passing through the point (0, —1, 3) and parallel to the plane 33: — y + 5 = 0. Hum coo. sum: is 7\ . (3,4,0) so CL): *51+CZ*A.=0 wi-m ctr—'5, 5‘4, c=0 (,u'uas 3"‘64450 3 nos 'm oo'm-r (0,4,1) 10 some coeJL 4+J.=o => &=~\ EQ'» 0F Punt is 3x:— "=0 ANS 3X‘3 -l=O (b) Find the distance between the points (1, —1, 1, 1) and (1,1,0,—1). (1"\x‘x‘)“ (l)\;o)"‘)= (0"7->\)1) 2c B\\$‘Y'ABL€ is l\(°,‘1,\’1)“- \i q+\+q = 3 ANS: 3 (c) Use projau = @1123 to ﬁnd the orthogonal projection of u = (1,3, —2) on a = (2,1,2). .a_ 1+3—q ( _ 7. \ z u. -__— 1.\,1)— ( ME (“44*“) ADAM/q) ANS; (WM) (d) Are the vectors (3, —1, 5,3) and (2, 4, —2,3) orthogonal? (‘5,-\,§,'b)—(1,H,~z,3)= CHM-‘0‘“) =4 7‘0 So Lb-r ounceeum ANS: U0 MATHl850/2050 809 Name: Page 2 of 7 (2) (4 marks) Find the steady state vector q for the following regular transition matrix 1/2 4/5 P = [1/2 1/5] up“)? 1 so qu-r 93:1 g,_e_ (1-9)jog ., L in I“? = ['/1. ’u/g‘l mow X ‘ X's/51° "V1— q/C (Lem-3“ O o 0 \$0 L€r 111—: S, y 1‘: 8k S ail-\g/Sls a. 14m: s=—‘— = E- Tued %;X1ISK=L=XV3\ is we. stew; swz Jet—Tot- (3) (5 marks) Find all the unit vectors orthogonal to V1 2 (1, —3, 2) and V2 2 (2, 1, 1) “\Ell‘: \i‘ISH-Nﬁ = {3—3 50 _)_C i; ”6? A on.“ JECTOL TAKE gt. = ELTHE ”é?“ Wﬁ. ﬁg?) 9 (W3i% s’x-fﬁ) UL, & 0L2 Att TUE OuL7 un‘w vecms L To \_’.\ 8‘!‘ MATH1850/2050 809 Name: Page 3 of 7 (4) (7 marks) The set S = {M1,M2,M3} Where 12 01 0—1 M1=i2oiM2=iuiM3=L1 oi is a basis for the vector space of 2 X 2 symmetric matrices. (a) If A = [ g _21 ] ﬁnd (A)g, the coordinate vector of A with respect to the basis 8. L€'\' A = L\r’\\+ LMﬁ‘LJ"; - WM“ \4\:\‘1x\‘3 (,e-v sys'rem 1;. = 2 23‘ +\¢L—\43 = 2 12‘ +kL-k3 :1 k1, = -\ w H'- H A uene mo MATlLix l O 0 7. O 0 Z ‘ QM o | O -\ k ‘4\--‘ 2-) LL=—\’ k3: \ 7. 1 -\ 'L ———> z ‘ _‘ 1. MOLE O O ‘ | o \ O —\ O 0 0 O (b) If (3)8 = (—1, 1, 2), then ﬁnd 3, ®)\$=('l,\,1) =) @=(_\‘)r’l‘*k_Mz+l}/\3= t; 'i] MATH1850/2050 809 Name: Page 4 of 7 (5) (7 marks) Let S = {V1,V2,V3,V4} Where v1 = (1, —2, 1),v2 = (—1,2,0),v3 = (—3,6, —3),v4 = (2,5,2). (a) Determine whether 8 spans R3. (Show all your work for full marks.) LET (a .lo, c) 2.5 (o [2,3 3“ \4\ ‘l\+‘41d_1*‘4393+\’-~(\JH‘(Q\B,L) GET sys'rm \L\ —\¢1, -3>\¢3-\ 3.ch CL ~2\;.«2\4t+ek,+; \L.‘ = ‘o \4 —?>\: -\-?_\L -'—c ‘ ‘ “ (we “9‘ " '3 7' Q 0.16921‘0—‘4 1 o” w \1 u now: m: b Mk‘ﬂz {x \ —\ —'5 -7, 7_ L and 5) o O \ ZqL-‘cké—aoGoo-q-I- \ o o —a.-\-<‘— s; THC \$Y\$T€H is Auwvis causisﬁm', Sp 3 SOAas W} (b) Is 8 a basis for R3? (Brieﬂy justify your answer.) . . . ‘ '5 M01 A Msxc ; S is Lu). savanna“, smce Any ‘1: dead“ m “2' A”: pagesSAcuuy So. 0 o o®%¢u;] MATHl850/2050 809 Name: Page 5 of 7 (6) (5 marks) Let p1 = 1 — a: — 2332, p2 = —2 + 23: + 5332, p3 = 3 — 4a: — 63:2. Determine Whether the set S 2 {p1, p2, p3} is linearly independent. (Show all steps for full marks.) LET \4‘ Qdkmu—hﬁs =0 ea 9mm k\ -2.l4,_-\-?>\4~5 :0 -\¢\ +9.21 4w} = o ’2\L\'\'S\41-L‘L3 =0 "'I 7.», wnu coat—Hutu? MA-nixx I -7. “a A: ‘7. g ‘1. TUE system “As om», mac vim)“ sewn, in: ”A750 M A = \‘o 1: ‘2:\V~2.'-Q1+lu= C—\)(—\)h3\\ '1": |+o -7.5 -L g‘L 5° TiNiAL \$0Lxﬁ|bo Ls uniaue , so SRs LG). ms. MATHl850/2050 809 Name: Page 6 of 7 (7) (9 marks) Let V 2 M33, the vector space of all 3 X 3 matrices with the usual addition and scalar multiplication, and let W denote the subset of all 3 X 3 diagonal matrices. (a) Prove that W is a subspace of V. cucoL “abuzz omen. Aoém‘on ._ \F A ,3 {n u) , can: At?) Cs {9W- A,(5muJ Mm FEELS] g: ‘23:; O Q} 3 l o oo‘0 1'th A-x-(S: \qﬁb‘ O O ‘\ is u.) UJ AS WELL. ‘33 0 AA“ 0 a. 0 Q3-\- cuccv. cLosuuz uncut. Scum. Muninlm-iou: it: A “\s {o UJ 3, k is AW P—k’ﬁk : CRELZ VA {“33“}. mm» A As ASP“? , lnA= ‘44, ° ° \ is {o UJ As we»... 3 o o la! \$0 \k) is A SO§¢A€E OF \I. (b) Exhibit a basis for W (you need not prove it is a basis). loo goo Can 000 o\o ooo coo ’ooo ’ col MATHl850/2050 809 Name: Page 7 of 7 (8) (5 marks) True / False. Indicate Whether the following statements are always True or sometimes False. (a) The plane 23: — 3y + 2 = 0 is a subspace of R3 True E/False (b)u><V=V><u True E/False (c) u is orthogonal to u >< V 3/ True False (d) If u is orthogonal to V + W, then u is orthogonal to V and to W True 3/ False (e) The set of polynomials of degree 2 or less with positive coefﬁcients is a subspace of P2 True EK False (f) The vectors (2, —4, 2, —4), (—3, 6, —3, 6) are linearly independent True 3/ False (g) No 3 vectors in R5 Will span the entire space 3/ True False (h) The orthogonal projection of u on a is orthogonal to a True Q/ False E/True False (j) If {V1, V2} is a linearly dependent set, then so is {V1, V ,Vg} True False (i)u><3u=0 ...
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