Show that B = {(1, 1, 1), (1, 1, 0), (0, 1, 1)} is a basis for R3. Find the coor

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let a(1,1,1) +b (1,1,0)+c(0,1,1)=0 => a+b=0 , a+b+c=0 and a+c=0 => a=b=c=0 hence {(1,1,1),(1,1,0), (0,1,1)} are 3 linearly independant vectors in R^3. since a set of any 3 linearly independant vectors in R^3 must form a basis of R^3, these vectors form a basis of R^3 now let a(1,1,1) +b (1,1,0)+c(0,1,1)=(1,2,3) => a+b= 1, a+b+c=2 and a+c= 3 => a=2, b=-1, c=1 hence the cordinate vector of (1,2,3) relative to B is (2,-1,1) feel free to ask doubts.. please rate and reward :)

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