Let \(T:\mathbb R^2\to\mathbb R^2\) be an isometry for the Euclidean metric. Prove that \[ T(x)=a+U(x) \] for some \(a\in\mathbb R^2\) and some orthogonal linear transformation \(U\).
Solution
Set \[ a\coloneqq T(0) \] and define \[ U(x)\coloneqq T(x)-a. \]
1. The map \(U:\RR^2\to\RR^2\) is an isometry satisfying\[ U(0)=0. \]
By definition, \[ U(0)=T(0)-a=0. \] For \(x,y\in\RR^2\), \[ \norm{U(x)-U(y)} = \norm{T(x)-T(y)} = \norm{x-y}, \] because \(T\) is an isometry.
2. The map \(U\) preserves the Euclidean inner product:\[ \inner{U(x)}{U(y)} = \inner{x}{y} \]for all \(x,y\in\RR^2\).
Step 1 gives \[ \norm{U(x)}=\norm{x}, \qquad \norm{U(y)}=\norm{y}, \qquad \norm{U(x)-U(y)}=\norm{x-y}. \] Using the real polarization identity, \[ 2\inner{u}{v} = \norm{u}^2+\norm{v}^2-\norm{u-v}^2, \] we obtain \[ \begin{aligned} 2\inner{U(x)}{U(y)} &= \norm{U(x)}^2+\norm{U(y)}^2-\norm{U(x)-U(y)}^2\\ &= \norm{x}^2+\norm{y}^2-\norm{x-y}^2\\ &= 2\inner{x}{y}. \end{aligned} \]
3. If \(e_1,e_2\) are the standard basis vectors and\[ u_j\coloneqq U(e_j), \]then \((u_1,u_2)\) is an orthonormal basis of \(\RR^2\).
By step 2, \[ \inner{u_i}{u_j} = \inner{e_i}{e_j} = \delta_{ij}. \] Thus \(u_1,u_2\) are orthonormal. Two orthonormal vectors in the two-dimensional space \(\RR^2\) form a basis.
4. For every\[ x=x_1e_1+x_2e_2, \]one has\[ U(x)=x_1u_1+x_2u_2. \]
By step 2, \[ \inner{U(x)}{u_j} = \inner{U(x)}{U(e_j)} = \inner{x}{e_j} = x_j \] for \(j=1,2\). Since \((u_1,u_2)\) is an orthonormal basis by step 3, the coordinates of \(U(x)\) in that basis are precisely these inner products. Hence \[ U(x)=x_1u_1+x_2u_2. \]
5. The map \(U\) is linear and orthogonal.
Step 4 expresses \(U\) as the linear map determined by \[ U(e_1)=u_1, \qquad U(e_2)=u_2. \] Thus \(U\) is linear. Its images of the standard orthonormal basis form the orthonormal basis \((u_1,u_2)\) from step 3, so \(U\) is an orthogonal linear transformation.
6. Therefore\[ \boxed{T(x)=a+U(x)} \]with \(a\in\RR^2\) and \(U\) orthogonal.
The definition of \(U\) gives \[ T(x)=a+U(x), \] and step 5 proves the required property of \(U\).
7. Q.E.D.
Step 6 is the desired representation.