# Consider the set S = {<1,-1,-2> , <a,0,-a> , <1,1,a-2>} For which values of "a" would S be a basis for R^3 ?

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Let us check if given vectors are Linearly independent

Let for some scalar x,y, and z

`x[1,1,-2]+y[a,0,-a]+z[1,1,a-2]=[0,0,0]`

`` `[[1,a,1],[1,0,-a],[-2,-a,a-2]][[x],[y],[z]]=[[0],[0],[0]]`

The rank of the cofficient matrix is 3 if

so vectors [1,1,-2],[a,0,-a],[1,1,a-2] are linearly independent if `a!=0,-2`.

It will be form basis of the S.