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Geometry of Linear Maps

  • James J. CallahanEmail author
Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

The geometric meaning of a linear function \(x \mapsto y = mx\) is simple and clear: it maps \(\mathbb{R}^1\)to itself, multiplying lengths by the factor m. As we show, linear maps \(M:\mathbb{R}^n\to\mathbb{R}^n\) also have their multiplication factors of various sorts, for any n > 1. In later chapters, these factors play a role in transforming the differentials in multiple integrals that is exactly like the role played by the multiplier φ'(s) in the transformation dx = φ'(s)ds in single-variable integrals.With this in mind, we take up the geometry of linear maps in the simplest case of two variables.

Keywords

Linear Subspace Maximal Rank Coordinate Change Positive Orientation Invariant Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer New York 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSmith CollegeNorthamptonUSA

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