If I have $2$ vectors in the plane, how can I find the matrix transformation that when applied to each vector, makes them unit vectors? I am looking for the $2 \times 2$ matrix A that satisfies
$ \mathbf{A}v_1 = \begin{bmatrix} 1\\ 0 \end{bmatrix} \ \ \ and \ \ \ \mathbf{A}v_2 = \begin{bmatrix} 0\\ 1 \end{bmatrix} $
Where $ v_1 $ and $ v_2 $ are the column vectors.
The specific problem I need to solve is to find the single transformation that maps $ \begin{bmatrix} 120\\ 60 \end{bmatrix} $ to $ \begin{bmatrix} 1\\ 0 \end{bmatrix} $ and $ \begin{bmatrix} 90\\ 30 \end{bmatrix} $ to $ \begin{bmatrix} 0\\ 1 \end{bmatrix} $ however an answer that helps me with the general solution would be appreciated too.
Note that $A$ is simply the inverse of $$\begin{bmatrix} 120&90\\ 60&30 \end{bmatrix}$$