Archetype A is a system of linear equations with coefficient matrix, \begin{equation*} A= \begin{bmatrix} 1 & -1 & 2\\ 2 & 1 & 1\\ 1 & 1 & 0 \end{bmatrix} \end{equation*}
Do the columns of this matrix form a linearly independent or dependent set? By Example S we know that $A$ is singular. According to the definition of nonsingular matrices, Definition NM, the homogeneous system $\homosystem{A}$ has infinitely many solutions. So by Theorem LIVHS, the columns of $A$ form a linearly dependent set.