Question 25: Prove Theorem 7. (Hint: For (a), compute \({\left\| {U{\mathop{\rm x}\nolimits} } \right\|^2}\), or prove (b) first.)

Theorem 7states that consider that \(U\) as an \(m \times n\) matrix with orthonormal columns and assume that x and y are in \({\mathbb{R}^n}\). Then,

1. \(\left\| {U{\bf{x}}} \right\| = \left\| {\bf{x}} \right\|\)
2. \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\bf{x}} \cdot {\bf{y}}\)
3. \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = 0\) such that if \({\bf{x}} \cdot {\bf{y}} = 0\).

Prove of \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\bf{x}} \cdot {\bf{y}}\) is shown below:

\(\begin{array}{c}\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\left( {U{\bf{x}}} \right)^T}{\left( {U{\bf{y}}} \right)^T}\\ = {{\bf{x}}^T}{U^T}U{\bf{y}}\\ = {{\bf{x}}^T}{\bf{y}}\\ = {\bf{x}} \cdot {\bf{y}}\end{array}\)

Here, \({U^T}U = I\).

If \({\bf{y}} = {\bf{x}}\) in part (b) then;

\(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{x}}} \right) = {\bf{x}} \cdot {\bf{x}}\)

It signifies part (a).

Part (c) immediately comes after part (a).

Thus, it is proved that \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\bf{x}} \cdot {\bf{y}}\).