- #1
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 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!
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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