Question

Help on this question of Linear Algebra, thanks.

Show that 1 1 B= 0 5 4 2 -5 5 9 8 -7 is a basis for W, where W= 2s – 5t 3r + 8 - 2t r - 4s + 37 -r + 2s ER:r,s,t ER

Answer 1

Page Sol here We al 28-5t 3r+ soat 48131 -rtas ERY, 8, 8, t ER 4 fb g 2 F is May 5 8 7 E Basis of (To show (1) LI (Linearly Independent) G g 2 f it 5 -7 0 4 -4 to 262 +9G so -54 +803 -26 +54 -73 02 G - Alat 20 26+ gezo 54 -54 +8(3 0 - 46 +54 - 73 fun by (تنا) اللہ (ا)

Page by Cių - gli) we e get 2C2t 93 g (c, - I (24 (g) - o 26 & 9th 9h-gc, +366-gozo or 3 8C₂ -99 by Sců we get 50-56 +86 - 8( a - 412 + (3) 20 59 - Sat 83 &C, t. 32₂ - 86 320 -3C, + 27 -0. (V) - by 3 (vi) 38C₂- weget. ga-3(-16, +2762) 20 ge, age, 384 81₂ -UB=0 or a u put in di we get put in o we get goo also satisfies non 4 20 (220 (iv) so that go implies Go abore uro, y 20, so B ù L.I.

Date Page (in B spam ie we show linear combination every every rector tul element of of ES op B. so let 28 st 30+8-2t be vector my W of 8-48 +3t -rt 28 Such that a a ty 2 of 2 D 4 1 V + 2 28-5t zrts-at & t3t -ay y +92 5a-syt 82 antry 7Z in comparing 2-st = a-dy + 2 3rtsat 2y + 97 10 r-45 +3t -rf 20 8-54 of 82 - 40 +54 - 72 an above also get here niy z

Date Page ① - 90 A we gel 384 9x z 3rt - ab g(20 - st) 38y -In 38 -17s + 43t (V) f cili 8 ue get -32+ 27y &- 4s 734 84 28 28-5t) -3n+2 n+27 8 - 208 + 43t vi by (V) 3 (vi) 38 y 9x - 3(-3n+27y) = 38-178 +421 - 3 (5-20578-31 Y 38y- gn tgn. 43s - 86€ -Bly -439 y = 438 - 864 -st24 put in (V) 38 (-8f2t)-gn 3r-ita + 434 383 + 76t gx= 3r-17st 4]f - 384 + 764 - 38t 178 - 434 gn -218-38 + 33t gn -215-30 +33h 9 -70-8f1lt 3

Date Page Comment ! every arbity vector vector of of wa (choosen) of the the form & we have shown every such vector is the linear combinatim ad the element t of R co 28-5t than Irts-24 8-40 +34 -rf 28 4 7s-rtllt 2 (car (-8f2A) + 9 5 7 -4 B span w so B is the Baris of w.