Which of the following is a basis for the subspace of R3 spa

Which of the following is a basis for the subspace of R^3 spanned by {v_1 = [1 2 1], v_2 = [1 0 2], v_3 = [3 1 0], v_4 = [1 1 -1]}? {v_1, v_2} {v_2, v_3} {v_1, v_2, v_3}, {v_1, v_2, v_3, v_4} None of the above

Solution

R3 has dimension 3 and we ahve 4 vectors so given set of vectors must be linearly dependent

Let us check if first three vectors ie v1,v2,v3 is a linearly independent set

av1+bv2+cv3=0 gives

a+b+3c=0

2a+c=0

a+2b=0

So, a=-2b,c=-2a=4b

a+b+3c=0

Substituting gives

-2b+b+4b=0

Hence,b=0 ie a=b=c=0

SO, v1,v2,v3 form a linearly independent set and must span the subspace

Hence correct answer is

C.