How to create a vector function (mapping)?

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It depends on what do you consider as a "mapping". The symbolic expression defined by

sage: var('t')
t
sage: x = vector([1,1])*exp(-t)

represents a (symbolic) function, which you can evaluate:

sage: x(t=13)
(e^(-13), e^(-13))

or derivate:

sage: x.derivative(t)
(-e^(-t), -e^(-t))

and so on.

If you want to define a python function, then you have to write something like:

sage: x = lambda t: (e^(-t), e^(-t))
sage: x(13)
(e^(-13), e^(-13))