A function is called surjective, or onto, if every element in the codomain has at least one preimage in the domain. This means that the function covers the entire codomain, ensuring that there are no 'gaps' in the output. Surjectivity plays a crucial role in understanding the properties of linear transformations and their matrix representations, as well as in studying quotient spaces and isomorphism theorems, where it helps determine whether certain mappings are comprehensive enough to create equivalences between structures.
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