Question

Help with linear combination

Answer 1

I have \[\Delta s^2=-(\Delta t)^2+(\Delta x)^2+(\Delta y)^2+(\Delta z)^2\] and I know that \((\Delta \bar{t}, \Delta \bar{x},\Delta \bar{y},\Delta \bar{z})\) are linear combinations of their unbarred counterparts in the expression. \[\Delta \bar{s}^2=-(\Delta \bar{t})^2+(\Delta \bar{x})^2+(\Delta \bar{y})^2+(\Delta \bar{z})^2\] My book says we can therefore write \[\bar{s}^2=\sum^{3}_{\alpha=0}\sum^{3}_{\beta=0}\mathbf{M}_{\alpha\beta}(\Delta x^{\alpha})(\Delta x^{\beta})\] for some numbers \(\{M_{\alpha\beta};\alpha,\beta=0,...,3\}\)

can someone explain this to me? it's been a long summer without math!