Suppose that \(\mathcal {B}_1=\left \{\vec {v}_1, \vec {v}_2\right \}\) is an ordered basis for some subspace \(V\) of \(\RR ^n\). Let \(\mathcal {B}_2=\left \{-\vec {v}_2, -2\vec {v}_1\right \}\). Verify that \(\mathcal {B}_2\) is also an ordered basis for \(V\).

Let \(\vec {w}\) be a vector in \(V\). If the coordinate vector for \(\vec {w}\) with respect to \(\mathcal {B}_1\) is \(\begin{bmatrix}2\\-1\end{bmatrix}\), find the coordinate vector for \(\vec {w}\) with respect to \(\mathcal {B}_2\).

\[\begin{bmatrix}\answer {1}\\\answer {-1}\end{bmatrix}\]