By now, you have probably already figured out how to determine if a linear transformation is injective, and what its kernel is. You may also now begin to understand why Sage calls the null space of a matrix a kernel. Here are two examples, first a reprise of Example NKAO.
So we have a concrete demonstration of one half of Theorem KILT. Here is the second example, a do-over for Example TKAP, but renamed as S.
And so we have a concrete demonstration of the other half of Theorem KILT.
Now that we have Theorem KPI, we can return to our discussion from Sage PI. The .preimage_representative() method of a linear transformation will give us a single element of the pre-image, with no other guarantee about the nature of that element. That is fine, since this is all Theorem KPI requires (in addition to the kernel). Remember that not every element of the codomain may have a nonempty pre-image (as indicated in the hypotheses of Theorem KPI). Here is an example using T from above, with a choice of a codomain element that has a nonempty pre-image.
Now the following will create random elements of the preimage of v, which can be verified by the test always returning True. Use the compute cell just below if you are curious what p looks like.
As suggested, some choices of v can lead to empty pre-images, in which case Theorem KPI does not even apply.
The situation is less interesting for an injective linear transformation. Still, pre-images may be empty, but when they are nonempty, they are just singletons (a single element) since the kernel is empty. So a repeat of the above example, with S rather than T, would not be very informative.