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m250-1test - 1(10 5 Determine t so that the vector v...

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Unformatted text preview: 1. (10+5) Determine t so that the vector v = (75,4,0) lies in the span, of v1 = (—1,1,1) and v2- — (2, —3, 1). (b) After having found It, express your answer as an explicit linear combination. tin—1.1 anal h 46d: c|v‘+c2_v1=\/. l U“ H .L‘ o -l -l:+L+ 1 ‘ 0 Jc .\ 2. '9 K3+3£1 -t+q. o "I o o thz 2. (15) Find a linearly independent set of vectors that has the same span as v1,v2, V3 where v1 = (.1, -—2,1), V2 2 (—2,4, —1), V3 = (—4, 8,—1). Ne maul 'ko So‘Qm C‘V, + cht+c3v$=o «Mi elm/LL— 5% W‘WW—m °\,‘z,°3- flue V V V3 M0. DAM Dug-‘1’ . C‘V|+C1 V,_+C%V :0 a_—:), 1 "2 "'1' a. o o *2 '+ 8 c2 3 O L I -\ «l 3‘ Rz—a RL-W—‘U R3 "7 R34“ ‘ ‘ 2 __ q. ‘ ‘2 " H o l 3 ———9 O O O R1623 O O O o l 3 - ,3c _. — L‘ :4 o C‘- — 3 C‘ 2C1 t3 =9 C = ’1‘?) c +- 3c3 =0 \ 3. (15) If possible determine 1" so that the displayed set S is a set of dependent vectors. S: {(1,2,—1), (2,1,—3), (—1,7,7~)}. v‘ v2. v3 CIV‘+L1V1+C5V3;0 . I 7- -l c. o z ‘ 1 C2 : O -l -3 ’5' c3 0 Its-elm“ Rafi-"R! I 1 _, g 7. —-I ___; o -l 3 O ‘3 ‘1 K‘QJSE" o —-l 4-" 0 -ul IL“ I '2. ~l RsfiRBtez O _| 3 Li} ¢-Lk:0 E) W W‘- 3% “t ”“2”?” —-cz_+3c3=o C1: 3C3, Cl: ~57? Sci.- Csz 7:21:3’ Clz—S UL.“ +3v + («l 1 0:0 l 2 So QM ”3""; W’S‘T‘M‘OQ Call/c1"?> NOT ALLZ—em 4,3 6:“ ,ullvgg W 4. (7+7)(a) Compute the inverse of the displayed matrix. (b) Express the displayed matrix as a product of elementary matrices. . O 2 —1 A: 1 -—1 2 2 —1 3 2. -—| 3 o o l l -‘ 2 O i O ‘ "I ——9 O l 7.. --l i 0 0 O ti2.92.3 2. R‘s—9R3 -232. 1 .\ z o I o o l al 0 -7. I o o l x“ q. -2. R2. ——-> [31+ 23 l ”I 0 — 2 -—7 H o ‘ o l 2 "‘ o o l l H '7— R‘ —-> KM; ‘ o o _‘ -.S' 3 o l O a 7- "'l ‘ ~7— 5. (14) Find the L, U decomposition of the displayed matrix: R1", R2“ 22' '. E. 1 _| 6.(10+5) For the matrix A displayed below, compute A3 — 5A2 + 9A — 4I (b) Is A invertible? You have to give a reason. Hint: You do not need to make a separate computation, use part (a). (a) Al— " 'L’ ‘ it 7 -L+ -7_ i l _\o -H 5— A3: 27 12. "'H .1 g 0 ulO —-\\ A3— sA‘HA-uz :17 I7. -\ '5’ + q ”I O “r o 13 a7 4 + o 4 2.0 4 -3f 2.0 —5' —S' O +4) Decide if the following statements are true or false. If you decide they are ample to indicate why. as a column of zeros then the matrix :10 = 0 has a non-trivial solution 7. ( 4+4 false give an ex (a) If a n x n matrix h (b) A is an n x n matrix. A a: = 0). Then A is invertible. (c) Given the columns of a 3 x 3 matrix are all distinct. The column vectors are linearly is not invertible. ( that is a solution other than 2- 9 3 i9. W V v2. v3 ...
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