show that x1123 x2101 x3567 form a linearly dependent set An

show that x1=(1,2,3) x2=(-1,0,1) x3=(5,6,7) form a linearly dependent set?

Answer should be 3x1-2x2=x3 but I do not know how to get this answer. Please help!!

Solution

Let ax1 + bx2 = x3

a(1,2,3) + b(-1,0,1) = (5,6,7)

(a - b , 2a , 3a + b) = (5 , 6 , 7)

Equating :

a - b = 5
2a = 6
3a + b = 7

From 2a = 6 ----> a = 3

a - b = 5
3 - b = 5
b = 3 - 5
b = -2

Confirming that thid works for the last equation :

3a + b = 7
3(3) + (-2) = 7
9 - 2 = 7
7 = 7

So, a = 3 , b = -2 works

So,

ax1 + bx2 = x3

becomes

3x1 - 2x2 = x3 ---> ANSWER