Proving the standard matrix U of T to be orthogonal

Question

So my class is getting into orthogonality, however, our reading assignments haven't been touching on transformations. I have this proof problem that I cannot seem to get around. Does anyone have any advice? Here is the problem:

Let $W \subset \mathbb{R}^{n}$ be a subspace. Consider a linear transformation $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ such that $T(\mathbf{x}) \cdot T(\mathbf{y}) = \mathbf{x} \cdot \mathbf{y}$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}$.

Prove that the standard matrix $U$ of $T$ is an orthogonal matrix.

Answer 1

BIG HINT

$$T(\vec{x})*T(\vec{y}) = (\vec{x})T^{T}T(\vec{y})$$

Answer 2

We have $$x^T U^T Uy = x^Ty$$ for all $x,y \in \mathbb{R}^n$. Now let $x=e_i$ and $y=e_j$, where $e_k$ denotes the vector zero at all locations except at $k$, where it is $1$. Now note that for all $i,j$, we have $$e_i^T U^T Ue_j = \sum_{k=1}^n U(k,i)U(k,j)$$ and $$e_i^Te_j = \delta_{ij}$$ Use this to conclude what you want.

Answer 3

One of the definitions of Orthogonal matrix is that the matrix sends orthogonal vectors to orthogonal vectors. You can show the matrix associated to $T$ does that by using that , for the standard basis $e_1,e_2,..,e_n$ , $T(e_i)T(e_j)=<e_i,e_j>=0$ , so orthogonal vectors are preserved ( using linearity of $T$ , the result extends.)

Answer 4

By the condition given, we have $\begin{align} (U\textbf{x})^{T}(U\textbf{y}) &= (\textbf{x}^{T}U^{T})(U\textbf{y}) \\&= \textbf{x}^{T}(U^{T}U)\textbf{y} \\&=\textbf{x}^{T}\textbf{y} \end{align}$ for all $\textbf{x}, \textbf{y} \in \mathbb{R}^{n}$, where the first equality is by the properties of transposes of products of matrices. But the last equality holds if and only if $U^{T}U = I_{n}$, so $U$ is an orthogonal matrix, as desired.

To prove this "if and only if" statement explicitly, it remains to show the "only if" direction. The standard matrix of $T$ has $j$-th column $T(\textbf{e}_{j})$ where $\{\textbf{e}_{1}, ..., \textbf{e}_{n}\}$ is the standard basis for $\mathbb{R}^{n}$. Now, consider arbitrary $\textbf{e}_{i}, \textbf{e}_{k} \in \{\textbf{e}_{1}, ..., \textbf{e}_{n}\}$. By the condition given, we have $\ T(\textbf{e}_{i}) \cdot T(\textbf{e}_{k}) = \textbf{e}_{i} \cdot \textbf{e}_{k}$, which is $0$ if $i \neq k$ or $1$ if $i = k$. Thus, we have shown that the columns of the standard matrix $U$ are orthonormal, proving that $U$ is an orthogonal matrix.