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Prescribe a map $F:\mathbb{R}^2\rightarrow\mathbb{R}^2.$ Define $F:=(x,y)↦(e^x,e^y).$ So $F$ takes points $p∈(x,y)$ as input and "exponentiates" them to yield new points $p′∈(e^x,e^y).$ Then the image of the nonlinear map acting on points in $\mathbb{R}^2,$ lives in the first quadrant of $\mathbb{R}^2.$ $F$ is easily seen to be a nonlinear map.

Sometimes it's useful to understand how mathematical objects change after being embedded in different spaces. Consider the transformation of the following algebraic curves under the mapping and how the equations change:

Take the equation of a circle in the real plane. $x^2+y^2=1.$ After the map it can easily be shown that the equation, embedded in our new "nonlinear space," is $\ln^2(x)+\ln^2(y)=1.$

Similarly, a rectangular hyperbola $xy=1,$ in our new space becomes $\ln(x)\ln(y)=1.$

$p(x)=x^n$ becomes $p^*(x)=\exp(\ln^n(x)).$

Identify a group structure in the image space for $x^2+y^2=1$ and $xy=1.$ Show whether the group structure in the pre-image space is isomorphic to the group structure in the image space.

I know that generally, if you have a group $G$ and a bijection $G→H,$ you can just push the group structure forward from $G$ to $H,$ which is precisely the group structure needed to turn the map into an isomorphism.

I think the group composition definition will have to be compatible with: $f(x \cdot y) = f(x) * f(y).$

KReiser
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J. Zimmerman
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1 Answers1

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Informally, exponential maps change addition into multiplication. This is based on the fact that $e^{x}e^{y} = e^{x+y}$. You want a group operation $\star$ on the image space of $\mathbb{R}^2$ under $F$, that is $\mathbb{R}^2_{\geq 0}$, such that $(x,y) \mapsto (e^x,e^y)$ is a homomorphism.

Hence, you want $F$ to respect the group structure $\star$: for any $(x_i,y_i) \in \mathbb{R}^2$ we need $$(e^{x_1+x_2},e^{y_1+y_2}) = F((x_1+x_2,y_1+y_2)) = F((x_1,y_1))\star F((x_2,y_2)) = (e^{x_1},e^{y_1}) \star (e^{x_2},e^{y_2})$$

And now you can see why it makes sense to define $\star$ as the multiplication.

For the yellow part of your question, first write down the group structure that you are putting on $x^2+y^2=1$ or $xy=1$ and then try to identify what it should become on the image space using a similar process. What happens?

AnalysisStudent0414
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