MATH 218 Exam 2 with Solution, Fall 2006

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 218 - Quiz IC Fall 2006-2007 Name... Ss odublon. seeveneene Section: 4 (@ 12:30) 5 (@ 9:30) Time: 60 min I- (a) (10 points) Determine whether the set W of all n x n matrices A such that AT =—-Aisa subspace of the set of all n x n matrices M,n with standard matrix addition and scalar multiptication. OD Suppose Aew 2 AT.-A onl BEW , QBT--B C A+B) = ATA QT. -A-B= — CA+8) so A+tRew (eAyTe RATS be (-Ad= - C KN) go kAEW muh ph'cahion , thot io Wiba ubrpacs ot a (b) (2 points) Is the set of all diagonal n x n matrices a subspace of W? No, Sine af A i, nxn AT mony be LL Fenant than -A , so Ha act of ald diagqenal nxn mature. ou net a nrubret hw. II- Find a basis for: (a) (7.5 points) The line 2x — y = 0 in R?. QXx- =O at d ye at teIR Ary pot om thes Lin has the fom (¢, 2t tt Therefore C20} 5 a bass fn He Line ony ee. (b) (7.5 points) The plane 2x — y = 0 in R? z=t yrs Ke Lb et¢ 2 2 Ang point an the, plane Roo the. for (44,2) = Cts, st )= s(S 0) 4 t (o,0,1) Moreen {(4,00), (2,0,} 2 Beccary 4x dependent Anes if Ke (LLpl+k looN~e tan tk =o So k= keo, k,=0 k,=0 IH- (10 points) Let v; = (1,2, 2,3), ve = (0,1, 0,0), v3 = (5, 12, 10, 15), vq = (—1,1,—-2, -3). Find a subset of {v,, v2, v3, vg } that is a basis for Span {v1, v2, Us, v4 }. Ae ft ° 5 -! > ! fo) Ss -f <R. 2 t 12, i oO | a 3 2 © Ilo -4@ S$ 08 15 -3 oo 2 8 (e) (2) oO fo) spaces of K, them fle: {aw bases “Iv. (a) (8 points) Given null; (A) - dim Cru space of A) (continue your answer here) % = X= Qt x= -at-4E =-64£ (4% )% 2 (64, 2t,£) =t(-6,2,!) Thine foe LCGrx~) 4 a bare, for the mills pnen of A. nell (A)=1 (b) (5 points) Define rank(A) then deduce it from part (a). 2 vank(A)= dim Ceolumn spon of A) =dm ( rw S prove 24 A) 2 vank (A) + nullity (A) = 3 C Dimension theoum ) so vank(A)= 3-422 (c) (5 points) Is it true that the row vectors of A do not span R°? Why or why not? Yes, since ml, (A) +o. By the egy valence. themsm , the Agw Veebos of A de net span R?, V- (10 points) Let T: R® —+ R‘ be a linear transformation such that T(1,2,0) = (3,1,—1,4), T(2,1,1) = (0, 2,1, —8), T(2,0,1) = (1,0, —2, 0). Find the standard matrix for T. T (12,2) = b, C,t,e)= e,+2e, So Tle)+2TC,) = b, Wieder re so 27S 47) Tle, =k (2,0,))2e 4 @, Te) +Ted=b, Sobre the syslem. i vi D b 2 | b a 2 #o { b 3 —_> \ 2 rs) b, Qo -3 | -2b+b oO —-H | ~2beb oo —.N i+) L 5 wn Qo Oo Ss ws __/ —> | 1 Oo Oo b-3b, ab, } 1 o b- b 0 O t |} Yb-sh-2b Tle)< b - bak = (3,)-1 4) ~a(o, 2, 1-8) 4 1,0,-20) = (4, -3, 7 7 yy 2) Tle = but ° om io so -f) ~ (2-20) - 4 % -3) Tle) = Yb -3b- nl Z. 3, 6, (2, - Yo) The standard maton fa T=f$ 1 27 4 — 3 (2 The prumous mitted io Re standard way fo such Gpections but Ren Hen ub a mud easier wou. Cood= (QLi)- (2,01) so Tle,ey= T(2,1))- THe) = Co, 2,1,-8)_ (1,0,-2,0) = (-1,23,-3) ° Chee= (12,0) ~ 2 Coe) TUeed= T1272) -a Cete))= T2z,0)-atlo),e) = (3,-4.4) -4 (-1,%3,-38) ~ (3,U-"4) 4 (4,-4,-6, 16) = (5, -3,-3, 20) (0, 0,1) = (2,9, ») - aC1,°,0°) Tlo,o, N= Taal) - 2TCIL90,°) = Cho,- y0) - 4 ( 3-3,-7, 22) = (l,0,-2,0) + Clo, 61, -4o). (-3, 6,12, -40) So tRe standad malrux T te CT- (te) Te Tle] = 5 -l “3 -3 Q 6 -7 3 ie VI- DO NOT PROVE ANY OF YOUR ANSWERS IN THIS QUESTION UNLESS YOU ARE ASKED TO DO SO. a- (27 points) Write TRUE or FALSE. 1- W is a subspace of a vector space V if and only if ku+ veV for all scalars k and vectors u and v in V. 3 a a, 2- If the row vectors and the column vectors of A are linearly pe: £ Cas independent then A is square. [ 3- If A is an invertible n x n matrix then A and AB have the same row space. “J” 4- In an underdetermined system Ax = b where A is m x n, the column vectors are linearly independent. [ 5- In an overdetermined system Az = b where A is m x n, the column vectors do not span R™, “77 6- The union of two subspaces of a vector space V is always a subspace of V. [= 7- If W is a subspace of a vector space V and S is a subspace of W, then S is a subspace of V. 8-.T : R? — R? defined by T(z, y) = (x?, 22 + 1) is a linear transformation. F 9- If A is a 4 x 5 matrix then the smallest value for nullity(A) is 1. TT b- (8 points) If A and B are two 7: x n matrices. Is rank(A + B) = rank(A) +.rank(B)? If yes, prove this. If no, give a counter example. False Tales. he | 9° | cheadly vonk(A)= 2 \ o 1 6 o 09 BL -l| © 7) abo rank (2) - 3 ie) fo} -| A+ Q- e +8 > ° - Se rank (A+B)= 0 ooaea The means Kat vank (A+R) vark(A) + rank(k) GOOD LUCK