To test vectors in $\mathbb R^3$, form the column matrix and solve $Ac=0$. For $v_1=(1,0,1)$, $v_2=(0,1,1)$, $v_3=(1,1,2)$, the relation $v_1+v_2-v_3=0$ proves dependence. For three columns, a nonzero determinant proves independence; for other list sizes use pivots or homogeneous solutions.
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