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.
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© 2010 Springer New York
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Callahan, J.J. (2010). Geometry of Linear Maps. In: Advanced Calculus. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7332-0_2
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DOI: https://doi.org/10.1007/978-1-4419-7332-0_2
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Online ISBN: 978-1-4419-7332-0
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