The aim of this exercise is to show that SP₂(R)acts transitively on R² \ {0}.

To show that the group of special linear transformations \( \text{SP}_2(\mathbb{R}) \) acts transitively on \( \mathbb{R}^2 \setminus \{0\} \), we need to demonstrate that for any two nonzero vectors \( \mathbf{v} = (x_1, y_1) \) and \( \mathbf{w} = (x_2, y_2) \) in \( \mathbb{R}^2 \), there exists an element \( A \) in \( \text{SP}_2(\mathbb{R}) \) such that \( A\mathbf{v} = \mathbf{w} \).

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