Math 54 - Spring 2000 - Ogus - Midterm 1

Math 54 - Spring 2000 - Ogus - Midterm 1 - Math-11am 51W...

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Unformatted text preview: Math-11am: 51W Prefeuor A. girl: SW. NW Midterm l N arne 1% TA ' The boxes ‘oelow are for your seorosI do not write in them! Write your solutions in the spaces profidecl after each problem. Ex- plain your reasoning in all eases: you may be graded on your explanations as “roll as on your answers. 1. Find X, if possible. If not, explain why not. (a) g )=x+(‘f ,, Lg "g (C) x Z ' (—1y374) : X1712)! : \ E 2. Consider the matrices: 1 —1 1 ~2 -—3 1 0 1 0 —3 1 O 1 —1 —4 ~ 0 l 0 1 —1 A I: 3: 2 —2 2 —3 —5 A O 0 0 1 1 3 —2 3 -—4 —9 0 0 0 0 O / Assuming these are row equivalent: (3) Find a basis for the row space of A from among the rows of A. J XV} i :4 fig L, j z u 4 § i” ’\ f‘ l: 3 J (b) Find a basis for the column space of A from among the colums of A. ., fi 3 a l“ 5‘ / ‘ 3., / ,t 1 i) w t (c) Find a basis for the null space of A. , J W \(Z XLt 4X5 ' f“! " L ‘l r? yi‘?q’lé”9 r1 1% 3; LI ' if k {‘2 Ya Y.’ygegxgm) vs g¢§t éé (d) What is the rank of A? rtifi>i tiéfiy Vifi}: 3‘ e» 3 NQEPE=L 3. In each of the following examples, you are given a sequence of vectors in a vector space V. Answer the question, explaining your answer clearly, using complete sentences. Full credit will not be given if you just answer yes or no. (a) Does the sequence ((2,2,—1,4),(1,7,3,2),(1,4,3,—1)), form a basis for the vector space V = R"? (b) Does the sequence (ac2 — 29: + 2, x2 + 2x, 9:2 —1,x2 - 311: + 5) form a basis for the space of polynomials of degree less than or equal to 2? i 5 s" ' l" fl l l l l l g L, fl ‘1. O '" _ W O Lt Z , 1 k N 1 5;) M “i i) l f l _ I ' .. A a, i * , 19/") , . ~ to «z’é’gj :0 “"H i l a. J (0) Suppose that (111, v2, v3, v4) spans the null space of a 5 by 7 matrix A of rank 4. Is the sequence (111,112, 113,114) linearly independent? “Av? 4. Let M22 denote the vector space of all 2 x 2 matrices, and if A 6 M22, let tr(A) denote the sum of the diagonal terms, and let W be the set of all A 6 M23 such that tr(A) : 0. Pi"? - ,_ )("L (offiltut’fid Alx\rfi,.flfl\m7h :0 Xl-ic) w: a 11. G25 OlquClm:0 r "\r ‘7‘” r : rt,“ - O O ‘1 ; \ @flflflwfl v“ lo rullfl? 3) Ll 7A 1 An 3 (A) I (b) Compute the dimension of W. (“W'O ‘Lz I, f .r i iii/’19 ‘v C 3*) an (*4 l (ii ’G; t v a,, d at tn waif (A141 l I ) with respect to your basis. " #1 6 1 ~. All” H (H (IV-hm v) 0 2 3 0 on .14? (c) Find the coordinates of < / MOA‘+-2A2<\%fi5 \ f /r:r‘/; \ \ \ Lat: 74; [I 1“ (Fl? GUM M W to '2 £741? J :3! / r,» <1 / ,/ I’ j ...
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This test prep was uploaded on 04/02/2008 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at Berkeley.

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Math 54 - Spring 2000 - Ogus - Midterm 1 - Math-11am 51W...

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