Question

basis for the row space of A and its Let A (a) Find dimension 1 1 2 12 o to -S (6) Find a basis for the column Space of A and Its dimensiun (c) Find a bars for the onell space of A and the A @ Find the rank op

Answer 1

Sounds given matrix Az 12 20 6 5 4 C 2 -2 1 04 20 -18 (a ) oppily (R3-9 281 +Re] 2 -2 ५ 0 20 -18 12 10-96 10 2 R2 Now, as S apply R R2 Ry 2 -2 5 -9 12.3 10 -96 02 R: 46,-782] 2 o 15 9/23 0 apply [R I R, 2 R2 0-354-2 o 15/3 o o o o 0-3 S -2 Lo 15. 9/2 3 0 0 Need dow space is a 6 0 -3 siz - 2 and 2] it's dimension is 245 IŚ -9123

- - 2 -2 20 -18 12 4 دم -g 6 ५ 2 -2 5 - 9/23 0 Lo o 2 10 G -g apply [R3 R3-2R 2 2 - 2 Pivot Column from Briginal (b) apply [Ro & R₂ + 2R, apply [Re P2 aply CR, 4R, -2 To obleen basis for the Calenn Sface we use the AN and dimension is XL O I 5 9/23 0 -28 ] 512 -2 O -3 ol 0 0 0 972 3 6 0

c) first we apply Cro R2+2 R, 2 -2 -18 12 4 20 -18 09 O TO -go apply [R2 I R 1 2 -2 los 15 -972 3 0 2 lo g 6. apply (R3 R3 -2 R2 3 I 2 -2 5 / 3 0 0 -2 apply (Road Riko 10- Siz - ó Now Convert to an equivalent System equivalent system and add an O 1 S 912 3 o o ecuation for lach free variblee น 1 + - 343 + 0 १५ + -235 = + 212 + 543 + -9/44 345 = 0

43 - 40 04 du 25 +6 Now Convert info vector भा 313 -SIA 245 -53 น 9/24 -345 + + A . ง ५ । A था Go 52 2 -5 का -3 + 4 o +4 ही है o o N hence, basis for hall space is = sly TER

(d) apply [Rg & Rg + 2 R1 12 -2 - 0 Y 20 -18 12 2 2 lo ag 6 - R₂ s als hun Uandi. Ver 2 - 2 a - 5 2 lo -yor 3 0 IL ALIHU IUP TUTT 6 . R3 I Rz-2R2 2 2 5 -495 3 0 hence, there remaining is 2. hon-reso rows.ro So Rank (A)=2