Who is a subject. Whom is an object. Swap in he or him: if him fits, use whom. This is pronoun case, not the contraction pair in who's versus whose. The broader rule is in pronoun case.
The swap
The student who finished early went to the library. He finished, so who is right. The coach whom we asked stayed late. We asked him, so whom is right. After a preposition, it is almost always whom: the teacher from whom I borrowed the book.
People, not things
Who and whom are for people. Things take that or which, covered in that versus who. The whole relative-pronoun set is in relative pronouns.
The bottom line
He = who. Him = whom. After to or from, start with whom.
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