Question

1. Consider the following two bases for R3. --{():(1) 0)) -= {(8) 0) (1) and (a) Compute Ps-B where S is the standard basis for R.

Answer 1

The given bases for Red are, and " Psal where s is the standard basis. S: [!] ro l lol lo To find PsB we write the elements of B in the linear combination of elements of s and find the scalars. lel a, b, c are the scalars, C1,-1,0) = a (1,0,0) + b(0,,) 4669,,1) (a,b,c). → a=1; b ), cio (Q, !, -0 = a C1,,0) + bíle) + 6(0,0,17 - a=2 b=l, c= -1 (Q10,0) = a (1,0,0) + b (9,4,097 000,0,1) . a=2 b=o, co

Since matrix Psy B is the of scalars. transpuékion of the 01. Il a 27 - 1 dis PB -8" . We write the elements of B' in the linear Combination of elements of B. (2,9,-1) = a (1,-1,0) +6 (2,1,-1) 70G, 2p) (2,0,-1) = Carabtac, - atb, -b) ☆ -6:-1, -2+6=0 atabtacto bel, -atles , 1+2+2c=2 by competing - a=-1 a = 1 26=2-3 20 = -1 c= 1 ( , , 3): a ( 1,-1, ) + b ( 2 ) -1) + (a - b) (3,1,3) = (a+abtac, -atb, -b) arabtac=0 . -atbal, b=3 - 6=-3 -4-6 +2c=o 26 =10 Co 5 -a-3=1 - -a=4 a=-4

(1,1,1) a (1,-1,0) + 6(2,1,-1) + 6(2,0,0) (1,1,1) = Carabtac, atb, -b) atab +2c=1, atb=1., b=1 - 2-2 +2c -a-1=1 b = -1 -a = 2 2c=lty a=-2 CE 562 Perleſ -17'ri -4 -2 L-2 - 5/2/ H2 5 512) i) Р р Similarly, (1,-1,0) = ac2,0, -1) +b co, 1,3)+((I)!!!) (1,4, 0) Qats, b+c, -a73b+c) 2a+c=1, b+c=1, - a +36+c=0 29=10 b -6, -(1-6) +36-1-0) 16:- azla -1+C 3 -30 20 2 a = 1 + 1} , b = -1 + 1}

(2,1,-1) = a(0,0,-1) +(, 1,3)+((1,1,D) (2,1,-1) = 2a + , b+c) - 2+3670) da+c=2 b+c=1 -4+36703-4 20.-2- c i bal-c, -(2-6) +367-6) +c =- a = 2-6 I b=1-2 ,-1+6 +3-3c + c = -1 :0 b--1 2-{:-/ -ZC=-1-2 C=-3x2=2 (2,0,0) = aCQ,0,-1) + b(0, 1,3) + C(1,1,1) Caje, od =(aa+c, btc, - @+36+C) da+cEd, b+c=0 -2+3b+c=0 2a-2-ci b=-6,-(2-4) -36+C =0 a = 2- ci b= 2 a = ñ 2+2 -1+&-&c=0 ca! a = also wie PB8 = 1 % 4/3 -713] [1% 0 8/6]