Question

show that the vectors a= (1,2,3) ,B= (2,-1,4)and C= (-1,8,1) are linearly dependent.
​

Answer 1

Answer:

Given a, b, c are linearly independent

∴pa+qb+rc=0⇒p=0,q=0,r=0...(1)

Now consider x(a−2b+c)+y(2a−b+c)+z(3a+b+2c)=0 or

(x+2y+3z)a+(−2x−y+z)b+(x+y+2z)c=0

Hence by (1) we have

x+2y+3z=0,−2x−y+z=0,x+y+2z=0 Above is a set of homegeneous equations.

Δ=

∣

∣

∣

∣

∣

∣

∣

∣

1

−2

1

2

−1

1

3

1

2

∣

∣

∣

∣

∣

∣

∣

∣

=−3+10−3=4



=0Since Δ



=0, the system of equations has only a trivial solution i.e., x=0,y=0,z=0 Hence linearly independent