Hallo,
a)
$$ \text{Lorentz-Trafo in x-Richtung: }\\t'=\gamma(t-\frac { v }{ c^2 }x), x'=\gamma(x-vt),y'=y,z'=z\\(ct')^2-x'^2=(c\gamma(t-\frac { v }{ c^2 }x))^2-\gamma^2(x-vt)^2\\=\gamma^2((ct-\frac { v }{ c }x)^2-(x-vt)^2)\\=\gamma^2(c^2t^2-2vtx+\frac { v^2 }{ c^2 }x^2-x^2+2vtx-v^2t^2)\\=\gamma^2(c^2t^2+\frac { v^2 }{ c^2 }x^2-x^2-v^2t^2)\\=\gamma^2(c^2t^2+(\frac { v^2 }{ c^2 }-1)x^2-v^2t^2)\\=\gamma^2(c^2t^2(1-\frac { v^2 }{ c^2 })+(\frac { v^2 }{ c^2 }-1)x^2)\\=\gamma^2(c^2t^2(1-\frac { v^2 }{ c^2 })-(1-\frac { v^2 }{ c^2 })x^2)\\=\gamma^2(\frac { c^2t^2 }{ \gamma^2 }-\frac { x^2 }{ \gamma^2 })\\=(ct)^2-x^2 $$
