$\{v_1+v_2,\, v_2-v_3,\, v_2+2v_3\}$ is a basis whenever $\{v_1,v_2,v_3\}$ is

Write \(w_1 = v_1+v_2\), \(w_2 = v_2-v_3\), \(w_3 = v_2+2v_3\). In the ordered basis \((v_1,v_2,v_3)\) these have coordinate matrix \[ A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & -1 & 2 \end{pmatrix}. \] A short expansion gives \(\det A = 3 \neq 0\), so \(A\) is invertible. Thus \((w_1,w_2,w_3)\) is obtained from a basis by an invertible linear change of coordinates, hence is itself a basis of \(V\).