Is an Orthonormal Set Necessarily Orthogonal?

[SeannyBoi71]

I know that an orthogonal set is not necessarily orthonormal, but here is my question: Is an orthonormal set necessarily orthogonal? I figured the answer would be yes, except on my last assignment I had a question that said blah blah if {u,v} is an orthonormal set blah blah. But my instructor gave me a hint that said make sure that u and -v are orthogonal. Are they not automatically orthogonal since they are orthonormal?

 

[micromass]

I know that an orthogonal set is not necessarily orthonormal, but here is my question: Is an orthonormal set necessarily orthogonal?

Yes.

I figured the answer would be yes, except on my last assignment I had a question that said blah blah if {u,v} is an orthonormal set blah blah. But my instructor gave me a hint that said make sure that u and -v are orthogonal. Are they not automatically orthogonal since they are orthonormal?

You misunderstood your instructor. You claimed that {u,v} is orthonormal. This is the same thing as claiming that {u,v} is orthogonal and that $\|u\|=\|v\|=1$. Your instructor wante you to explicitely check orthogonality, because he wasn't convinced that you knew that it was orthogonal.