Math 2270 Assignment 6 - Dylan Zwick Fall 2012 Section 3.3 1, 3, 17, 19, 22 Section 3.4 1, 4, 5, 6, 18 - - 3.3 The Rank and Row Reduced Form - 3.3.1 Which of these rules gives a correct definition of the rank of A? (a) The number of nonzero rows in R. (b) The number of columns minus the total number of rows. Jhe number of columns minus the number of free columns. (d) The number of l’s in the matrix R. 04 i (c) 1 c Qt 3.3.3 Find the reduced R for each of these (block) matrices: (0 0 O A=(003) 2 4 6) A A 7 c=(AO B=(AA) h rc-l &Vt€ [c’a6 b (003 / y3: O0 rou 8 00 rd) b z 7 rc I / 3 Cc] ( t3 Oo( ( /o3 I “ I 5bfrec- 3,’ 2 ocj cølVc ) I OCH C) Oo\ oa3 BQidec’( Oou 00 G1 000 Oo$ 0o3 0C) C) Ccc) / v_o /(Z3 001 00 / Ool iLo Qe 2 GC) 1l oaf 00 Oo( 6 C2d OC)C) Co / / 3.3.17 (a) Suppose column j of B is a combination of previous columns of B. Show that column j of AB is the same combination of previ ous columns of AB. Then AB cannot have new pivot columns, so rank(AB) <rank(B). 1 and A 2 so that rank(A B) 1 (b) Find A B=( , = 1 and rank A B 2 = 0 for ). A - - ( Ab, t - - ( -- b) 1 A AL 1o Uc2 () 1(1 c{flk 0 7 / 3 I. 3.3.19 (Important) Suppose A and B are ii by n matrices, and AB Prove from rank(AB) < rank(A) that the rank of A is n. So A is in vertible and B must be its two-sided inverse (section 2.5). Therefore BA I (which is not so obvious!). = rk k (A ) fr (4) rck(A) Arc k 4) 4 ) [r ()Z 3.3.22 Express A and then B as the sum of two rank one matrices: rank=2 4— / ( /1 A=f 1 1 O’ 14) 1 8) I B= 2 2 23 /0 0 l!O)4/o() 1)J /0 0 5 ‘e Complete Solution to Ax = b. 3.4.1 (Recommended) Execute the six steps of Worked Example 3.4 A to describe the column space and nullspace of A and the complete so lution Ax = b: /246 4\ A( 25761 1 /b 2 bf b /L C A J] - c/ 6 ‘1 4 i 3 1° zq Vi o 9zJ 5 7h 2 LI Y 4 =1 3 \s ) 3 \\b \2 3 5 2) I /4 )=r( C/ 6 (f I I 0 0 0 4i),’-2 A 6 ( 5, ?4 — Co IP ipe 1 A 4Il (a2 cO)p1I1_ C( yD) / (I)/ @) () 6 4 O I C) I I 1 0 0 ( C) - / I I L o°OdzJ -‘-ZbL 1 -b, 1 h bLJ-?b, )/oi 1 ( 1 0 ( ; (h200 ? 1 1 C- (40! ‘ / PS J /Pooç 7Yx /(h 200) JI h2 1 ji7J\ Qc i; () “Tv j f)Q( l2Lj J /i hOQ hZoo ) 1 I — I /T) \\1h11 o (T79) I 1/ oT (uojnos jviaua q; pajj osi) uoT4nJos a4aJdwoD aq puj 3 is the system solvable? Include 3.4.5 Under what condition on b ,b 1 ,b 2 b as a fourth column in elimination. Find all solution when that condition holds: z 5 ç Lf + 2y + 41 + 5 y y 9 — — — h( -L -L/ x 2x 6 2z = 4z 8z = 62 = 63 / z-z -i0 10 1(3 /IL JO 7(oic Oo - L/ /V/ 2, /o_z 1P4 (3/0 0 o / b, L 1 b? /•I ( fJ 8 / 0 C)loJ /1 11 Y c. ) KL 0 — r —c \L N nJrJ \[) I) A — - r- I— I I \ 0 N ) --,p —, c-J f Th L’3 — — .. cJ I) — — cDc.)1kD t’3 I n 0 —, — — — C ) U U — = —__ — — 0 — — —cr 0 1J I 1 CD I II I’ 0 CD C C z C CD c,3