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I am having problem trying to prove the following:

Let S and T be bases for a vector space V , and P be the transition matrix from S to T. Suppose that S is orthonormal.

Please prove that T is an orthonormal basis if P is an orthogonal matrix.

After some research online, all I could find was the proof of the converse: if S is an orthornomal basis, then the transition matrix from S to T is orthogonal.

Nevertheless, here is my attempt:

$$ \text{Let } S = \{ u_1 \cdots u_k \}, T = \{ v_1 \cdots v_k \} $$ $$ \text{By defintion, } (u_1 \cdots u_k ) = (v_1 \cdots v_k)P $$

If P is orthogonal, then $P^{T}$ is also orthogonal because column space of P form an orthonormal basis for $ \mathbb{R^n} $.

$\therefore$ By defintion of orthogonal, $ P^{T} = P^{-1} $. Hence $ P ^{-1} $ is also orthogonal.

Thus, $(u_1 \cdots u_k )P^{-1} = (v_1 \cdots v_k)$

I think that I just re-proved the converse of the problem. I can't see how to complete the proof in the other direction. Could someone please advise me?

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To make notation simpler, let $\mathcal{S}$ and $\mathcal{T}$ be the bases. Form matrices $S$ and $T$ from the respective basis vectors.

Since $P$ is an orthogonal matrix, we have $$PP^T = P^T P =I.$$

Since $\mathcal{S}$ is an orthonormal basis, $S$ is an orthogonal matrix and $$S S^T = S^T S = I.$$

Using that $P$ is the transition matrix from $S$ to $T$, \begin{align*} T &= PS \\ T^T T &= T^T PS = (PS)^T PS \\ T^T T &= S^T P^T P S = S^T S = I. \end{align*} Similarly, \begin{align*} T &= PS \\ TT^T &= PST^T = PS (PS)^T\\ TT^T &= P SS^T P^T = P^T P = I. \end{align*} Since $T^T T = T T^T = I$, we have that $T$ is an orthogonal matrix. Since $T$ was formed from the basis vectors of $\mathcal{T}$, we have that $\mathcal{T}$ is an orthonormal basis.

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