Question

Consider an ellipse Ein ${\mathit{R}}^{\mathbf{2}}$centered at the origin. Show that there is an inner product $<.,.>$in ${\mathit{R}}^{\mathbf{2}}$ such that E consists of all vectors $\overrightarrow{\mathbf{x}}$with $\mathbf{\left|}\mathbf{\right|}\overrightarrow{\mathbf{x}}\mathbf{\left|}\mathbf{\right|}\mathbf{=}\mathbf{1}$, where the norm is taken with respect to the inner product $\mathbf{<}\mathbf{.}\mathbf{,}\mathbf{.}\mathbf{>}$.

Short answer

It is proved that there is an inner product in$R^2$such that E consists of all vectors$\overrightarrow{x}$with $\left|\right|\overrightarrow{x}\left|\right|=1$, where the norm is taken with respect to the inner product .

Step-by-step solution

Step by step solution Step 1: An inner product R2

Let $E=\left\{\left[\begin{array}{c}x\\ y\end{array}\right]\in R^2:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\right\}$

We will define a liner mapping which send ellipse to a circle. Define the linear mapping
$T:R^2\to R^2,\left[\begin{array}{c}x\\ y\end{array}\right]\mapsto \left[\begin{array}{c}x/a\\ y/b\end{array}\right]$

Observe that, T is a linear mapping and T maps ellipse to the unit circle.

Ker$T=\left\{\overrightarrow{0}\right\}$

Proof

Now, an inner product on $R^2$:

From the exercise 5.5.17:

Hence, the norm is taken with respect to the inner product is

role="math" localid="1659610758022" $T:R^2\to R^2,\left[\begin{array}{c}x\\ y\end{array}\right]\mapsto \left[\begin{array}{c}\frac{x}{a}\\ \frac{x}{b}\end{array}\right]$

It is proved that there is an inner productin$R^2$such that E consists of all vectors$\overrightarrow{x}$withrole="math" localid="1659610974581" $\left|\right|\overrightarrow{x}\left|\right|=1$ , where the norm is taken with respect to the inner product .