Answer with explanation:
Let, A=[1,1,2]
B=[1,2,1]
C=[2,1,5]
⇒Now, Writing vector , A in terms of Linear combination of C and B
A=x B +y C
⇒[1,1,2]=x× [1,2,1] + y×[2,1,5]
1.→1 = x +2 y
2.→ 1=2 x +y
3.→ 2= x+ 5 y
Equation 3 - Equation 1
→3 y=1
So, Vector A , can be written as Linear Combination of B and C.
⇒Now, Writing vector , B in terms of Linear combination of A and C
Now, let, B = p A+q C
→[1,2,1]=p× [1,1,2] +q ×[2,1,5]
4.→1= p +2 q
5.→2=p +q
6.→1=2 p +5 q
Equation 5 - Equation 4
-q =1
q= -1
→2= p -1
→p=2+1
→p=3
So, Vector B , can be written as Linear Combination of A and C.
⇒Now, Writing vector , C in terms of Linear combination of A and B
C=m A + n B
[2,1,5] = m×[1,1,2] + n× [1,2,1]
7.→2= m+n
8.→1=m +2 n
9.→5=2 m + n
Equation 8 - Equation 7
n= -1
→m+ (-1)=2
→m=2+1
→m=3
So, Vector C , can be written as Linear Combination of A and B.
So, All the three vectors , A=[1,1,2],B=[1,2,1],C=[2,1,5] can be written as Linear combination of each other.
⇒≡But , the two vectors, (0.4,3.7,-1.5) (0.2,0),can't be written as Linear combination of each other as first vector is of order, 1×3, and second is of order, 1×2.
