The Wedge Product and the Definition of the Determinant

This article uses the question in the title to motivate the wedge product, also called the exterior product. Many things that are traditionally expressed using the cross product in three-dimensional space have natural generalizations to any number of dimensions when expressed in terms of the wedge product. The wedge product can also be used to define the determinant of a linear transformation. Basic properties of the determinant are obvious from this definition.

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2025-01-26 (added two references about determinants, and added references to two newer articles in this series)
2024-05-21 (changed the title to be more descriptive, added a phrase to the abstract)
2024-02-25 (fixed incorrect grammar: "whether or not" -> "whether" when the outcome is in question)
2022-06-11 (updated references to other articles in this series to link to html abstract pages instead of to pdfs. Didn't change the version number/date)
2022-02-06 (reformatted title)
2022-02-05 (fixed pdf metadata)
2022-01-16 (first version)

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updated 2025-02-22

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The Wedge Product and the Definition of the Determinant

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