The vectors v1 v2 v3 are linearly independent Determine if t

The vectors v_1 v_2, v_3 are linearly independent. Determine if the following vectors are also linearly independent. the vectors v_1 - v_2, v_2 - v_3, v_3 - v_1, the vectors v_1 - v_2, 2(v_2 - v_3), 3(v_3 - v_1), the vectors v_1 - v_2, 2v_2 + v_3, 2v_1 + 3v_3.

Solution

a)

(v1-v2)+(v2-v3)+(v3-v1)=0

Hence they are not linearly independent

b)

6(v1-v2)+3(2(v2-v3))+2(3(v3-v1))=0

Hence they are not linearly independent

c)

Let, a,b,c so that

a(v1-v2)+b(2v2+v3)+c(2v1+3v3)=0

(a+2c)v1+(-a+2b)v2+(b+3c)v3=0

Hence, a=-2c,a=2b ,b=-3c

a=-2c ,a=2b gives b=-c

Hence, b=c=0=a

HEnce vectors are linearly independent