Question

Use the fact that matrices A and B are row-equivalent. A = 1 2 1 0 0 2 5 1 1 0 3 7 2 2 -2 10 23 7 -2 10 1 0 3 0-4 0 1 -1 0 2 0 0 0 1 -2 0 0 0 0 0 B = (a) Find the rank and nullity of A. rank nullity (b) Find a basis for the nullspace of A. (c) Find a basis for the row space of A. {95353 (d) Find a basis for the column space of A.

Answer 1

2 1 0 0 A- 25 - - do 10 3 0-4 0 1 -1 2 O BE 37 2 - 2 lo 23 2 ملم -2 10 o 2 I OO 3 25 37 -2 2-2 10 23 7-2 10 find bases for the row space, column the space and reduce a to an we need nuspaces an echelon form Re: R₂-28;, lz: Rz- 3Rq' Rus Ru- lor, 2 1 o 1 -1 1 0 0 -1 2-2 3 -3 -2 10 Ruu Ry-3R2 R3 R3-R₂ i 2 1 0 0 1 - 1 -2 6 0 O 0 -5 10

2 1 0 0 - -1 0 I-2 10 R: Rut5R3 I 2 - 1 4 o 1 2 نا 0 0 Ranc 3 Row .: Basis for (། ༢ » -- 0, 0) Row of = co -1, 1 (0,0,0 1,-2 ciel cd A Basis for 2 2 Cod A- 2 7 lo 23 Dimension of col A = 3 n Rank & dim (nula) dime NWA) 5- 3 dim (nula)=2

By eqo - - 1-1 2 R,: R, 2r, ' R : R2-R3 3 O- 20 - ~ - 1 2 2 с lo 3 - 2 0 1 -1 2 2 - 0 0 R,: RitzR3 1 0 3 - 4 0 2 - B 0 0 of= the equation Äă =ó is equalant that is -40g x, +393 4205=o "-" nur 2&s=o Hly - 2x5

and %-d3 + 2x,- =0 x2+2x= 3 and 2, + 3( 22 +225)– 4%- x,+ 382 + 6 xs - 4x = 0 =0 2, + 372 +2x, a -1322 + 2x;-) , - (3^2+285) av X2 x3 xy "2 trag 2x5 xs -3 as X₂ t 2 ༠༠ D for Nul A= Basis O dim (Nula 1