What is Einstein's principle of relativity?

Show that a spherical shell expanding at the speed of light, $\mathbf{x}^{2}=c^{2} t^{2}$, in one coordinate system $(t, \mathbf{x})$, is still spherical in a second coordinate system $\left(t^{\prime}, \mathbf{x}^{\prime}\right)$ defined by

$$\begin{aligned} c t^{\prime} &=\gamma\left(c t-\frac{u}{c} x\right) \\ x^{\prime} &=\gamma(x-u t) \\ y^{\prime} &=y \\ z^{\prime} &=z \end{aligned}$$

where $\gamma=\left(1-u^{2} / c^{2}\right)^{-\frac{1}{2}}$. Why do these equations define a Lorentz transformation?