Images of the unit circle under linear transformations of $\mathbb{R}^2$

Let \(T:\mathbb R^2\to\mathbb R^2\) have singular-value decomposition \[ T=U\begin{pmatrix}\sigma_1&0\\0&\sigma_2\end{pmatrix}V^T, \qquad \sigma_1\ge\sigma_2\ge0, \] with \(U,V\) orthogonal. Since \(V^T\) preserves the unit circle, \[ T(S^1)=U\{(\sigma_1\cos\theta,\sigma_2\sin\theta):0\le\theta<2\pi\}. \] Thus:

• if \(\sigma_2>0\), the image is an ellipse centered at the origin, with semiaxes \(\sigma_1,\sigma_2\); it is a circle when \(\sigma_1=\sigma_2\);
• if \(\sigma_1>0\) and \(\sigma_2=0\), the image is a closed line segment centered at the origin, of length \(2\sigma_1\);
• if \(\sigma_1=\sigma_2=0\), the image is the single point \(\{0\}\).

These are exactly the possible images.