I had thought I understood everything about orthonormality until I encountered the notation on the source I share below. Why add another step using T? In other sources, it worked just fine without. Could you explain the reason why a human being would spend extra bits and energy to add that?
For two column vectors $\vec x$ and $\vec y$ of the same dimension, we have $$\vec x\cdot \vec y=\vec x^T\vec y$$ based on the definition of matrix multiplication and the dot product. The transpose notation is more preferable for someone who exclusively wants to use matrix operations instead of vector operations like dot products.