Abstract
System (6.4) has k degrees of freedom if there is a set of k of the variables that can be freely chosen such that the remaining n − k variables are uniquely determined when the k variables have been assigned specific values. If the variables are restricted to vary in a set S in ℝn, the system has k degrees of freedom in S.
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References
For (6.1)–(6.16) and (6.22)–(6.25), see e.g. Rudin (1982), Marsden and Hoffman (1993) or Sydsæter et al. (2005). For (6.17)–(6.21) see Parthasarathy (1983). For Brouwer’s and Kakutani’s fixed point theorems, see Nikaido (1970) or Scarf (1973). For Tarski’s fixed point theorem and related material, see Sundaram (1996). (6.36)–(6.38) are standard results in linear algebra, see e.g. Fraleigh and Beauregard (1995), Lang (1987) or Sydsæter et al. (2005).
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© 2010 Springer-Verlag Berlin Heidelberg
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Sydsæter, K., Strøm, A., Berck, P. (2010). Systems of equations. In: Economists’ Mathematical Manual. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-28518-2_6
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DOI: https://doi.org/10.1007/978-3-540-28518-2_6
Publisher Name: Springer, Berlin, Heidelberg
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