Linear independence of orthogonal and orthonormal sets?

[Riemannliness]

(Note: this isn't a homework question, I'm reviewing and I think the textbook is wrong.)

I'm working through the Gram-Schmidt process in my textbook, and at the end of every chapter it starts the problem set with a series of true or false questions. One statement is:

-Every orthogonal set is linearly independent. ->My answer:True; Text: False

What's the deal? I thought orthogonality => linear independence. I know if the statement was the other way around then it would be false, since Linear independence =/> orthogonality.
I'd usually write it off as a typo, but the next statement is:

-Every orthonormal set is linearly independent,

which is true in my opinion and the text's, and that makes me think that there's a distinction being pointed out between orthogonal sets and orthonormal sets that I've missed.

 

[radou]

Take an orthogonal set of vectors. Add the zero vector to it. What happens?

 

[Riemannliness]

Oh snap! Good one.

 

[Fredrik]

Does the book's definition of orthogonal sets allow the 0 vector to be a member?

 

[Riemannliness]

Yes, the book takes the stance that the zero vector is orthogonal to every vector.

 

[radou]

http://mathworld.wolfram.com/OrthogonalSet.html

There don't seem to be any restrictions related to the zero-vector.

 

[gtse]

Need some clarification myself as well:

An orthogonal set is not always linearly independent because you could have a 0 vector in it, which would make the set dependent.

But an orthonormal set must contain vectors that are all orthogonal to each other AND have length of 1, which the 0 vector would not satisfy.

Is that the right logic?

 

[HallsofIvy]

Yes.

 

[Dosmascerveza]

Or perhaps you could argue that every orthonormal set contains vectors which are orthogonal with each other and this set is also a basis. Every basis is linearly independent. ==> every orthonormal set is L.I.