exam1_key - Math 304 Name: K E 10/4/2010 Section: Your...

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Unformatted text preview: Math 304 Name: K E 10/4/2010 Section: Your Score a In order to get credit, you must show all work. 0 Read all problems first. Try working on them from easiest to hardest. o No calculators of any kind! No cell phones! . . 1 0 ——1 —1 1 0 1. Con81der the matrices A — (2 0 1 > and B w ( 1 1 4). For each of the following operations, either do the indicated calculations or explain Why it is not defined. ‘ (a) 3A+2B (b) A-B (c) B-A (d) A-BT (e) A-BT—(B.AT)T (f) (A-AT)2 ‘ ~~3©"3 'Z.2_o_ k+2___3 Ox 5 + - " W O AZB léoglllllgl [s 2H] NOT DeFlN‘ED‘. #ag murj MA l?) i #0? colucmms {m CC) NOT DEFINED 5 flageolumms l“ “##6; VOL/U3 h/\' (90 ~ \ o ~—| ~\ \’ i-(—0+o~me\)~o (-l+0—l+tl)"l : "l “3} A0: 2’ 0 1} \ l 3 erwiflo z~\+”o-\+l-LI *1 Q o ‘l , “ T A V T_ 2. A @DAgdfiNY:A§?QOW§:A§FA «y; —0 cl 6>®Nlillifllgt 2. Given the following system of linear equations, Where t is some constant: x1+2w2+m3+x4+m5=1 x3 — 934 + m5 = —1 x1 + 21102 +’ 01133 + 2564 + 0335 = t. (a) Determine the augmented matrix of the system and identify the coefficient matrix. Then bring the associated augmented matrix of the system into row echelon form, indicating all elementary row operations. . l 2— l l l Qa£~r\ll \l vi?qu M5 64‘ oolrl 1~|-———>10©\-—\\'\ “(all I! ool‘ll’l 00630th (b) Give an explicit value for t such that the system is inconsistent. Then determine a value if for which the system is consistent. (SasJCew/x is consisiiavfli' ‘ufl owxcl mild Cl: t’llo. So “ml is Cans/{smut 0M6 ¥or til. It is Magus/Eleni %rwmplelrggrt10, (c) Using the value for t which yields a consistent system, determine the reduced row echelon form for the system. Indicate all elementary row operations. ‘lZ\\\\r..>u_g\202.02_ o0\-\\-\-A> oo\~\\*-\ 000000 0063000 (d) Identify the free and the basic variables of the system. Then determine the general solution of the s stem and check your result. (PW “W (€33 X1, XL] zyf ‘/ BUS/k, VatVfableS; XUXS , sanction: chat X‘: 7.”27<7." Z—Xq (1*2‘flz’2/5Lh‘t Zfzfel “N‘s” XSB txufl 7(5‘ 1 l \/ 7%“:qu «bx 'Xs CH “WEB” Y” “5‘24 V _ ~— Ll ~ — _ h— it: XL, (2 2x1 ng5+sz+ 2m! - L \/ 13 110 —1 3.LetA=(2 7) andB——<2 0 2 1). (a) Find A‘1 and check your result. 13 \l 0 garb—2V1- i3! \ O neg-3‘9, \ 0‘1} P3 21 o i O \ _—-> W Al: [1?] W M“: ('2: [EH ‘6] v 0 (b) Use your work from part (a) to express A—1 and then A as a product of elementary matrices. Then check your result. it “alarm? ii: = T . 36 ,_.. L’K‘J ~ [Chick A; [HEX MA fill; ii:- [L :1 «and» (c) Solve the matrix equation AX = B by using A‘1 from part (a). (You must use A‘l, not any other method.) ,4 ,‘1'3 \lO'l__ [tin-é “‘0 Y'AB‘ ’1l3[zo?~l‘[ A : i :1 x1+m2+w3 4. LetF 2 = $2+m4 . .171 +373 "—174 (a) What is the domain of F and what is the codomain? 3 (DOMOCLVI ‘. KL. Cfiéfimcu‘th IR l (b) Determine F(ei), z' = 1, 2, 3,4, for the standard basis {61, e2,e3,e4} of R4 written as column vectors. i [$330]: 1t 0‘ \+O‘O L C) o \ "i (c) Using (b), exhibit two vectors v,w E R4 with F(v) = F(w) but v yé w. Is F one—to—one? Explain. 1 (:0? V3€4 MA I 01' 33 gag—CM‘HOV‘) F (5 mi‘ Owl~+1>~oqu\ (d) Using (b), write down the standard matrix A of F such that F(x) 2 Ax. WHO 0’02 (OI-l (e) Determine the rank of A and show F is not onto. .\ \ s 0 WET \\ IO (ganja \ \_ \O anA22<$ oi©(-5‘O|ox—~—?O@OC\D gomnkvéeies: lO‘—‘ “ "i 000 HAMHULW‘ (Fe O \O wasmsi: is MJ!‘ Cid—0‘ (f) Exhibit a vector w E R3 which is not in the image of F. r . O a \\\Oq .a_lil@aYb-ar+fii\ Ol©ibILg>nOlOtb 3>@\O\bb ‘ ‘ C-0 O O O O Q’OW $6 Ckng VQdYDY M‘H‘ Q'OUrLfiEO noi‘ [:2 [k W mete 0i F For were [rifle EMF. . Circle “True” or “False” for each of the following problems. Circle “True” only if the statement is always true. No explanation necessary. ’34.. (a) n m X n matrix has m columns and 71 rows. v A,. (b)‘ If a 6 X 8 matrix A has rank 4, then any row echelon form has two zero rows. (0 Suppose that A is a 5 X 6 matrix. Then Ax = 0 has infinitely many solutions. as V (d) Let A be an m x m matrix. Then Ax = b has a solution x for any b E Rm. ' ' AV (6 ‘ The rank of a 12 x 6 matrix is less than or equal to 6. (f) , he row echelon form of an 8 X 5 matrix contains at least 5 zero rows. (g) The matrix 3) i (2) is in reduced row echelon form. A 118 r . 2 2 1 1 . . (h ‘ The matrlx [O 0 3 0] is in row echelon form. (1 Every elementary matrix is nonsingular, 0 Let AT - B be defined. Then (ATB)T : BT . A. ...
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exam1_key - Math 304 Name: K E 10/4/2010 Section: Your...

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