Sample Test

Sample Test - \1 also» sowmps MATH205O F08 V1 Name: (509,0...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: \1 also» sowmps MATH205O F08 V1 Name: (509,0 Page 1 of 8 (1) (4 marks) Give an equation of the plane containing the point P(2, 0, 1) and the line l : (:13, y, z) = (1, —1, 1) + t(1,0, —1) _.| - mule matched “EL-rot. Fol LNG ’1‘: (1,0,4) ' Pct-$0 am: 1 Nuts m um; (15,”) a (“A”); (,a— sacomb UELTOL L—r'w-Tomlu) In (1-\,o—c-n,\—\3= 0 mo) _\ —$ . . ' up 1"“ '5‘:J\x11=\ “ A k\= (1)")l) lo-l l\o «in 4‘ (bum. in R'CX—x.,7—Y.,2_-7_.) :0 «$5. (\,—I,\)'Cx«l,-3-O,1-\)=° of. x—~3+-2_—?>=o (2) (Markov Chains - 5 marks) Find the steady state vector q for the fol- lowing regular transition matrix: _P:: [ 1/3 3/5] 2/3 2/5 it i1- 14): 2/5 '3/5 Ran) ; ‘ 'q/io ° M 11=$)1‘=%5 -273 3/5 IGBOLE o a o MATH205O F08 V1 Name: Page 2 of 8 (3) (6 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 — $1 -- $2 -- 23173 102 = —$1 -- [€362 -- 23173 103 — $1 -- + 2)$2 -- + 1)$3 Know T a. 4‘4 Mfflato, 0_P~ Thigh—4 Miflw. \ I ‘L l \ Z \ \ 7. M Pa = ~\ \4 2 = 0 kn Li film = 0 km ‘t l \<. \-'L km 0 k H \c-\ K; Kl O 0 11-5 = Q4*‘)(\4’5\ =0 L=~\ A ch So T991, 331% H ks-lm has. (b) (2 marks) Let T : R2 —> R2 be the one-to-one linear transformation defined by T(£U1, $2) = ($1 — $2, 53171 — Find the standard matrix [T4] for the inverse transformation. UH; —\ _ "‘_ _,\_ -5 ‘ ’ ‘5/1 yz [Ti-[fl—mst A-[WZ \] MATH205O F08 V1 Name: Page 3 of 8 (4) (5 marks) Let p1 = 1 + 33,192 = 1 — 332,193 2 1 + 333,194 2 3:3. Determine whether the set S = {p1,p2,p3,p4} is a basis of P3, the vector space of polynomials of degree 3 or less with real number coefficients and the usual addition and scalar multiplication. l l l 0 K o -k __ O o 'l o O \‘3 o I k : o 0 ° ° ‘ k‘ 0 k _ W CALL. THE “EFF. mm A M W Lhofl‘aQIw-QMM'AIQKEEI. 1 cl: 3! \ (co mam. a9. ALom. «W 2m») 0 -\ o MA= —_ @yL-qan‘xl A: Z #0 g - - ak=L=L —«O xMuMAq’O z 3 a Y1 So sam.g& 3. MM“ m93=m—.q,m&swu.m¢f “egg—magma. $0 $5AMsis MATH205O F08 V1 Name: Page 4 of 8 (5) (5 marks) Let V = P3 (the vector space of all polynomials of degree 3 or less with real coefficients and the usual addition and scalar multiplication), and let W be the set of all polynomials in P3 of type p = a + b3: + (a — b)a:2 — a333 where a, b E R. (i) Show that W is a subspace of V. - wean CLosuua UMBEL fiofii'riofi : p.16 Kl ii.» @166. $00! v.16 ‘THEQ e: gala; +(q—lo\ x} ~Q><3 5;: Q’ +i<>ix-\-(Ql—‘o'\7<t ‘o‘lxs “Ll-Iv q;‘°)°~,)(°,em- “men 9+1: (GA-0.0+ C‘in)x -\- [CK-Fm *CL+‘J)]xI—(q+q’)x3 6K1) . CHECK unsung (moan. 50¢“ Mum—m. -. % few} and half?!“ H06“). éAy 94v: dram 8. hell. k9 = 0m.) -\- (tux {(kaA-(kbflxl—QQW evd 36‘ U) (A Q. We. 4% (ii) Give a basis of W, and state the dimension of W A lama. {Mutt—x} , x—xus ohm) = .2, MATH205O F08 V1 Name: Page 5 of 8 1—1 0 1 0 0 (6)(5marks)LetM1=[ 0 0], M2=[ 0 2], M3=[ 0 1]. The set S = {M1, M2, M3} is a basis of the vector space of 2 X 2 upper tri- angular matrices. fi)H(A);=(21,—D,fimiA. L a o l J . Find (B)g, the coordinate vector of B with A= U’l\+\”\z’\”\5= 1 1 0 2 respect to this basis. Lt Q): \<\M‘+\L1Hz+\‘3M3 (8» Cg)5=(1<nl‘nl‘5)) L& m = @LaB=[ MATH205O F08 V1 Name: Page 6 of 8 (7) (5 marks) so X» 52., x.‘ is 243 7 —2 @2G09—10 1 2 1 2 1 00 3 0 LetA— 122 5 1 andR—TreflA): 000 0G 1 20 —1 1 000 0 0 1~ 4~ «- Ir + 1‘ (a) Give a basis for the column space of A A Mme 1s i[1\ \ 111-,[3 \ 2, 01T,[-7.\ \ (31.1] (b) Give a basis for the nullspace of A M X1: 5, Xq=9§ NRA“. s‘ké. K j y): xc=o, xsz-bflu X\='1$'\'t so x, ~2. ‘ XL 1 O *3 1 0 5+ '5 It x.1 o l x.5 o o & so QKWQMWGAGA {[410 o o3T,E\ o 3 \ ong (C) State the following: (i) TCLTLk<A> 3 (ii) nullity(A) = l, MATH205O F08 vl Name: Page 7 of 8 (8) (5 marks) True / False. Indicate whether the following statements are always True or sometimes False. (a) Elementary row operations do not change the column space of a matrix True if False (b) In R3 rotations are one-to-one linear operators 3’ True False (c) If T is a linear transformation, then T(u + kv) = T(u) + kT(V) 3/ True False (d) The plane 23: — y + 3,2 — 2 = 0 is a subspace of 3-space True Z/ False (e) The projection of the vector u on the vector a 7é 0 is orthogonal to a True Z) False (f) For any m X 71 matrix A, rank(A) + nullity(AT) = m K True False (g) The set of all n X n matrices with even entries is a subspace of ]\47m True E/False (h) The identity matrix In is a unit vector with respect to the inner product defined by (U, V) = tr(VTU) True E/ False (i) In an inner product space, (ku, kv) = k<u,v> for any 11,V,]€ True E, False (j) If {V1, V2,V3} is a linearly independent set, then so is {V1,V2} True False MATH205O F08 V1 Name: Page 8 of 8 BONUS (5 marks) Let T := R2 —> R2 be a linear operator, such that T(2, 1) = (1, —1) and T(1, 1) = (0,2). Find [T], the standard matrix of T. BY '\‘\\A («3.3 [3]: [T350 \Tcezfl e. a 0,0) , aficon ’T C. UME‘AL km» T(2\\)=’\_(&e\+€z\ lYexfliezv U r‘) T (1:13: Tee'd'e'l) a Téexfipfifi-J = (0‘2) 50 Te.» = 0,-0 —<o,z)= ( l r3) Tum T(e,) —_ (0,1) _T(e.)= (on) —C\ :5) = (“315) ...
View Full Document

Page1 / 8

Sample Test - \1 also» sowmps MATH205O F08 V1 Name: (509,0...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online