Write each vector as a linear combination of the vectors in

Write each vector as a linear combination of the vectors in S.

Write each vector as a linear combination of the vectors in S. (Use s_1 and s_2, respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.) S= {(1, 2, -2), (2, -1, 1)}

Solution

a).

z=(-9,-3,3)

s1=(1,2,-2) s2=(2,-1,1)

z= as1 +bs2 ( a,b any numbers)

(-9,-3,3) = a(1,2,-2) + b(2,-1,1)

(-9,-3,3) =(a+2b , 2a-b , -2a+b)

a+2b =-9 2a -b =-3

after solving we get a=-3 and b=-3

so z = -3s1 -3s2

b)similerly for (-2,-6,6)

(a+2b , 2a-b , -2a+b) = (-2,-6,6)

a+2b=-2 2a-b =-6

solving we get a=-14/5 b= 2/5

v =-14/5s1 +2/5 s2

c). w=(1,-18,18)

(a+2b , 2a-b , -2a+b) =(1,-18,18)

a+2b=1 2a-b =-18

a=-7 b=4

w = -7s1 +4s2

d). u=(3,-5,-5)

(a+2b , 2a-b , -2a+b) = (3,-5,-5)

we can\'t write it in linear becasue

2a -b = -5 and -2a+b =-5