This could perhaps be very silly but I am confused and I need help. Consider
Write both of the following 1-parameter groups of transformations as a composition of commutative 1-parameter transformations: $$ \psi_1^s\left(x_1, x_2, x_3\right)=\left(x_1+s, x_2+2 s, x_3+3 s\right), \quad \psi_2^s\left(x_1, x_2, x_3\right)=\left(e^s x_1, e^{2 s} x_2, e^{3 s} x_3\right) $$ Hence write down the vector fields which generate these transformations, and check your answers are correct by showing the relevant ODEs are satisfied.
I don't understand what they mean by the composition of commutative 1-parameter transformations. For example would just doing component-wise operators work? I don't know if this is allowed as this will make the whole thing trivial. If someone can clarify this for me I would really appriciaite it.
As it is worded, the considered transformations may be decomposed into several combinations of one-parameter transformations, but given that those transformations have to commute, the natural choice leads to component-wise transformations indeed, so that $$ \begin{cases} \psi_1^s = \phi_1^s \circ \phi_2^{2s} \circ \phi_3^{3s} \\ \psi_2^s = \chi_1^s \circ \chi_2^{2s} \circ \chi_3^{3s} \end{cases} $$ where $\phi_i^s = e^{s\partial_{x_i}}$ translates and $\chi_i^s = e^{sx_i\partial_{x_i}}$ dilates the $i^\mathrm{th}$ coordinates respectively.