Solution to Fall Midterm II

# Solution to Fall Midterm II - MATHl850/2050 809 Name Page 1...

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Unformatted text preview: MATHl850/2050 809 Name: Page 1 of 7 (1) (5 marks) Find an equation of the plane containing the point P(1, 1, 1) and parallel to the two lines given by the parametric equations BlEGCTlOL) “Exam [1: 3321—75, y=2+2t, z=3 j|=(—1,2,0) l2: 3:23—2t, y=4+t, z=t ZEfC—AIM) c. 5 \<\=(1)\33) -—\ L o "7— ( l & 0st qx-l‘o 8+6: +oL=O W1“ C4 5’6) =1) = C2. ‘99 3° Zx+I%-\’?>7_-\-0L=O s; Mal. x=\,\Q=/l,z:zt To GET," 1+ \+2,+at=o =9 A: —L. SO iln‘k‘a-k'B—L—Q'JO (2) (5 marks) Say T : R3 —> R3 is the linear operator deﬁned by: a reﬂection about the asy-plane followed by a counter-clockwise rotation by an angle of 7T/ 2 about the positive QU-aXlS followed by an othogonal projection onto the yz-plane. Use [T] = [T(el) | T(€2) | T(63)] to get the standard matrix of T. Z 2 z MATH1850/2050 809 Name: Page 2 of 7 (3) (7 marks) (a) (4 marks) Let T : R3 —> R3 be the linear operator deﬁned below. De— termine the values of k for which T w to be one-to—one. 101 — 2:171 -- \$2 -- 2:173 102 = —2£E1 -- [€362 -- 2:173 103 — 2:171 -- + 2)\$2 -- + 1)£U3 T mks To rag A—A @ MET—3:0, Z l 2, ’2. \ 1 —1 l; 1 = Q \m u QZ+L= 2 tan \m 0 b1 \c'\ 1341‘ 'L l 2. Mfr}: 0 Lu L+\ :2 00k.“ -91 3 = loankxX =o \L=—\ oil. \u: 30 T llo 103‘: 4'4 « \LL—rk 0L \(,=§ (b) (3 marks) Let T : R2 —> R2 be the one-to—one linear transformation deﬁned by T(\$1,£E2) = 23171 — \$2, 53171 — Find the standard matrix [T4] for the inverse transformation. m= [; ii if} ﬁat—13s A: / MATH1850/2050 809 Name: Page 3 of 7 (4) (5 marks) Determine whether the set S 2 {p1, p2, pg, [94} is linearly depen- dent / independent, where p1=1—3327192:_1+\$+\$27p3=1+\$—\$27p4=_1+\$2+\$3 L€T \¢\9\+LLQI-\i¢5\>3+l¢qpq =0 96 )<",><‘)>c1)><S 0" 3‘57“ 5‘55 “LS/“g \L\_'\(_1 +iég,’\4q : O —k\ +i¢z~ \LéA—kt‘ = O \Cq = O \ ._.\ \ .( O \ "\ \ “l O o l l o 0 o, \ \ o o _l \ -\ \ 0 22:134.; 0 o o o o O O o l 0 o o a \ o l —\ \ -l 0 _’——E 0 \ i O O 2.544 LR o O O ‘ 0 o o O o O' 3‘“ [L3 Ls A new. vmmsl’i, we (a «so/MM? scurrth 5 [#3 VMmuLAz, "P-‘U‘AL Sim“ ‘5 pd? upxékf.) s: S is Lin- mm —. MATHl850/2050 809 Name: Page 4 of 7 (5) (6 marks) Let V = F(_OO7OO) be the vector space of all real-valued functions with the addition deﬁned by (f+g) = f(a:)+g(a:) and scalar multiplication deﬁned by = k(f(a:)), and let W be the set of all functions f in F(_OO7OO) such that f(a:) = a1 sin(a:) + a2 cos(a:) where a1, a2 6 R. (i) Show that W is a subspace of V. 6) case! cLosoug mum. mica-cu -. {ﬁe UQ )swud Sci-K €,%€ so gm): <>H111I~60 +qz ceoCx) L gcx) = \mmeH (mood) “zit/ﬂ, ﬂ.)‘l.1;l°nl°zéM-* TM @400): {€ngch (Ql+t,)ﬂaao<>+ (quanta L/xr—’ (/w—J €12 6 “2. so {JV 6%). 3 @ 0&ch (,LosolLE ()me SucLAfL MULTkOLtCI/l’ﬂéubz. geuj & Le [K’sﬁow LIX/6w. Lei—Hr} Q As A&°\J€- , Nd'b leéWL. My” =k~£0<>= (bgmoidiﬁym so \élgew~ 61L é “L § LL) Us A sowst—oe 01% (ii) Give a basis of W, and state the dimension of W A Mia-6 l\$ 22 M34176, mxll v 59v- MATH1850/2050 809 Name: Page 5 of 7 (6) (7 marks) Xx xuxb x11 15 243 7 —2 6320—10 1 2 1 2 1 00®3 0 WA: 2 44 10 2 andRZWﬂA): 000 06) 360 —3 3 000 0 0 (a) Let S = {V1,V2,V3,V4,V5} where v1 = (2,1,2,3),v2 = (4,2,4,6),v3 = (3,1,4,0),v4 = (7,2,10,—3),v5 = (—2,1,2,3). Give a basis for span(S) A 2""615 ‘6 lg\)\J-7J>\l§\l (b) Give a basis for the nullspace of A XrLgX'K Owl/'60., @0— XLE\$)YR:£ “£04. \$,%em Tues) R3 , x5 :0 L\ X\= —l.><-'L_\')<'q : g X, "Z. \ a O X x; 2 s4, ‘3 £ 3) A- wts FOL ‘5 (c) State the following: (1) Tank(A) = 3 (11) nullity(A) = 1 3 (111) Tank(AT) (iv) nullity(AT) = \ MATHl850/2050 809 Name: Page 6 of 7 (7) (5 marks) True / False. Indicate whether the following statements are always True or sometimes False. (a) Elementary row operations do not change the row space of a matrix Er True False (b) The rank of a matrix A is equal to the number of parameters in the solution of Ax = 0 True 3/ False (c) If T is a linear transformation, then T(0) = 0 E/True False (d) The plane 233—y+3z— 2 = 0 is parallel to the line (:13, y, z) = (—1,1,1)7L t(0,1,1) True E/False (e) The vector (1/\/§, —1/\/§,0) is a unit vector with respect to the Eu- clidean inner product 3/ True False (f) For any m X n matrix A, nullity(A) is at most the minimum of m and n True j/False (g) The set of all polynomials of degree 2 or less with even coefﬁcients is a subspace of P2 True E/False (h) If 81 2 {61,192,193} and 82 = {61, 61—1—62, 61+63} then spa (81) = span(82) True False (i) If W is a subspace of P7,, then dim(W) is at most n True E/False (j) If {V1, V2,V3} is a linearly independent set, then so is V1,V2} True False MATHl850/2050 809 Name: Page 7 of 7 BONUS (4 marks) Let T1, T2 be two one-to—one, linear operators on R". Prove that the composition operator T 2 T1 0 T2 is one-to—one as well. T 5T1 4" M W => [Tiiill Wm WM Mm— m [TX ° ETC“,sz K \M-UELﬂ—k-JLE AS weLL (Pmoa— a? when-Isa mules) so T 15 4*! ...
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