Is it true $GS(TA)=T GS(A)$ where GS denotes the Gram Schmidt process and $T \in O(n,\mathbb R)$?

Question

Let $A$ denote an $n \times n$ matrix.Suppose $GS:GL(n,\mathbb R) \to O(n,\mathbb R)$ denotes the Gram Schmidt orthogonalization process.Is it true that

GS(TA)= TGS(A) where $T$ is an orthogonal matrix?

I think the claim is true but unable to write a proper proof.Any ideas?