Every vector $\mathbf{v}$ can be expressed uniquely in the form $\mathbf{a} + \mathbf{b},$ where $\mathbf{a}$ is a scalar multiple of $\begin{pmatrix} 2 \\ -1 \end{pmatrix},$ and $\mathbf{b}$ is a scalar multiple of $\begin{pmatrix} 3 \\ 1 \end{pmatrix}.$ Find the matrix $\mathbf{P}$ such that
\[\mathbf{P} \mathbf{v} = \mathbf{a}\]for all vectors $\mathbf{v}.$

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