Vector Space Question-1

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*1/3 =-1

⇨ a₂₌ _-2/3

Putting a₂& a₃ in equa(2)

We get a₁=2-2/3 -2/3

⇨ a₁=2

Since from (a)

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

Vinay Kumar Mishra said...: This comment has been removed by the author.; 26 August 2013 at 00:35
Anonymous said...: Thank you so much; 26 August 2013 at 00:36

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