Show that the most general Lorentz transformation can be written as a \(4 \times 4\) matrix \(\hat{\Lambda}\) satisfying

\[\hat{\boldsymbol{\Lambda}}^{\mathrm{T}} \cdot \hat{\eta} \cdot \hat{\boldsymbol{\Lambda}}=\hat{\eta} \quad \text { and } \quad|\hat{\boldsymbol{\Lambda}}|=1\]

Since a Lorentz transformation is by definition a linear transformation of time and space that preserves the speed of light, you simply need to show that these two properties as necessary and sufficient for this. Note also that reflections get ruled out by the second condition by choice.