Linear Combination-Example

Question:- Express (1,2,3) as a linear combination of (1,1,1),(2,-1,1) and (1,-2,5) in V₃(R).

Solution:-Let a_1,a₂,a₃∊R such that

(1,2,3)=a₁(1,1,1)+a₂(2,-1,1)+a₃(1,-2,5)……………………(a)

(1,2,3)=(a_1,a_1,a₁)+(2a₂,-a₂,a₂)+(a₃,-2a₃,5a₃)

(1,2,3)=(a₁+2a₂+a₃, a₁-a₂-2a₃, a₁+a₂+5a₃)

⇨                           a₁+2a₂+a₃=1 ……………………………(1)

                             a₁+a₂+a₃=2  ……………………………..(2)

and                       a₁+a₂+5a₃=3……………………………..(3)

Substracting equa(3) from (1)

We get                  3a₂+3a₃=-1……………………………….(4)

Again, substracting equa(3) from (2)

We get                -2a₂-7a₃=-1

or                           2a₂+7a₃=1………………………………..(5)

Now, multiplying by 2 in (4) & 3 in (5) and then substracting

(5) from (4)

We get                      15a₃=5

⇨                                    a₃=1/3

Putting ‘a₃’ in (4)

We get                 3a₂+3 =-1

⇨                               a_2=-2/3 

Putting a₂ & a₃ in equa(2)

We get                 a₁=2-  -  

⇨                          a₁=2

Since from (a)

(1,2,3)=2(1,1,1)-2/3(2,-1,1)+1/3(1,-2,5)