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etotheipi

I got a bit confused, and hoped someone could clarify a few things. As far as I am aware, a change of basis is an identity transformation ##I_V## on the vector space (pg. 113) and we can write the relationship between the components of some vector ##v## in the different bases ##\beta## and ##\beta'## in matrix form, like$$[\mathbf{v}]^{\beta'} = [I_V]_{\beta}^{\beta'} [\mathbf{v}]^{\beta}$$where the matrix representation of the identity transformation ##[I_V]_{\beta}^{\beta'}## is known as the change of basis matrix.

For a Galilean transformation ##G##, between two given coordinate systems, with matrix representation ##G(R, \mathbf{v}, \mathbf{a}, b)## where ##R## is the rotation transformation, ##\mathbf{v}## is the relative velocity, ##\mathbf{a}## is a translation, ##b## is a time boost, we can write the matrix form of the transformation like $$(\mathbf{x}', t', 1)^T = \begin{pmatrix}

R & \mathbf{v} & \mathbf{a}\\

0 & 1 & b \\

0 & 0 & 1

\end{pmatrix} (\mathbf{x}, t, 1)^T

$$I had a few questions about this. Firstly, the elements of the vector space on which the Galilean transformations act look like vectors with coordinate matrices in the form ##(\mathbf{x}, t, 1)^T##; what exactly are these vectors (e.g. is the vector space on which the Galilean transformation acts the space of 'events', or something?). Are ##(\mathbf{x}', t', 1)^T## and ##(\mathbf{x}, t, 1)^T## two different coordinate forms of the

If that's sort of along the right lines, does the same apply for the Lorentz transformations ##\Lambda## (i.e. would Lorentz transformations be identity linear transformations on the space of events?). That would again make sense, because a Lorentz transformation amounts to a re-labelling of the coordinates between inertial frames, but we're still referring to the

Sorry if I made a mistake... thanks!

For a Galilean transformation ##G##, between two given coordinate systems, with matrix representation ##G(R, \mathbf{v}, \mathbf{a}, b)## where ##R## is the rotation transformation, ##\mathbf{v}## is the relative velocity, ##\mathbf{a}## is a translation, ##b## is a time boost, we can write the matrix form of the transformation like $$(\mathbf{x}', t', 1)^T = \begin{pmatrix}

R & \mathbf{v} & \mathbf{a}\\

0 & 1 & b \\

0 & 0 & 1

\end{pmatrix} (\mathbf{x}, t, 1)^T

$$I had a few questions about this. Firstly, the elements of the vector space on which the Galilean transformations act look like vectors with coordinate matrices in the form ##(\mathbf{x}, t, 1)^T##; what exactly are these vectors (e.g. is the vector space on which the Galilean transformation acts the space of 'events', or something?). Are ##(\mathbf{x}', t', 1)^T## and ##(\mathbf{x}, t, 1)^T## two different coordinate forms of the

*same*vector in this underlying space, in which case the Galilean transformation is an*identity*linear transformation on this underlying space of events (i.e. similar in concept to the transformation ##I_V## in the example above, whose matrix representation is ##[I_V]_{\beta}^{\beta'}##)?If that's sort of along the right lines, does the same apply for the Lorentz transformations ##\Lambda## (i.e. would Lorentz transformations be identity linear transformations on the space of events?). That would again make sense, because a Lorentz transformation amounts to a re-labelling of the coordinates between inertial frames, but we're still referring to the

*same*event at the end of the day...Sorry if I made a mistake... thanks!

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