Prove that S 1 1 1 1 2 3 2 1 1 is a basis for R3SolutionLe

Prove that S = {(1, 1, 1), (1, 2, 3), (2, - 1, 1)} is a basis for R^3.

Solution

Let, a,b,c so that

a(1,1,1)+b(1,2,3)+c(2,-1,1)=0

a+b+2c=0

a+2b-c=0

a+3b+c=0

Adding last two equations gives

2a+5b=0

Adding twice second equation to first gives

3a+5b=0

Hence, a=b=c=0

HEnce S is a linearly independent set.

A linearly independent set of size 3 in R3 , which has dimension 3, must form basis for R3