Invertible Non-Linear Transformation

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SUMMARY

The discussion centers on the invertibility of the non-linear transformation defined by the equations \(y_{1} = x_{1}^{3}\) and \(y_{2} = x_{2}\). The participant expresses confusion regarding the concept of invertibility in non-linear transformations, contrasting it with linear transformations. They seek assistance in determining the invertibility and finding the inverse, while also mentioning the relevance of Jacobians and the inverse function theorem in this context.

PREREQUISITES
  • Understanding of non-linear transformations in mathematics
  • Familiarity with the concept of invertibility in linear transformations
  • Knowledge of Jacobians and their role in transformations
  • Basic principles of the inverse function theorem
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  • Study the properties of non-linear transformations and their invertibility
  • Learn how to compute Jacobians for multi-variable functions
  • Explore the inverse function theorem and its applications
  • Practice finding inverses of non-linear functions through specific examples
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Students and professionals in mathematics, particularly those studying calculus and linear algebra, as well as anyone interested in understanding non-linear transformations and their properties.

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Homework Statement


Is this invertible? If so, find the inverse.

\left( \begin{array}{ccc}<br /> y_{1} \\<br /> y_{2} \end{array} \right) = \left( \begin{array}{ccc}<br /> x_{1}^{3} \\<br /> x_{2} \end{array} \right)

Homework Equations


The Attempt at a Solution



I know that an invertible LINEAR transformation is a L.T. (linear transformation) that can be reversed (right?), so that if L : Rn -> Rn, M : Rn -> Rn, M \circ L results in the identity matrix of n.

However, I don't really know how to go about finding this, and besides, it is noted that this is a NON-linear transformation (now I'm really clueless). I also don't know how to prove this is invertible without just showing an inverse as my proof. Any help would be great.

Thanks.
 
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this can be thought of as two separate single variable functions if it helps as
y1=y1(x1) and y2=y2(x2)

also do you know about jacobians and the inverse function theorem?
 

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