Linear independence of orthogonal and orthonormal sets?

In summary, the conversation discusses the relationship between orthogonal sets and linear independence. While it is commonly believed that orthogonality implies linear independence, the textbook presents a counterexample by stating that not all orthogonal sets are linearly independent. However, the conversation also recognizes that every orthonormal set is indeed linearly independent. The distinction between the two types of sets is further explored through an example involving the zero vector. It is concluded that an orthonormal set must contain vectors that are both orthogonal and have a length of 1, while an orthogonal set may have a 0 vector which would make the set dependent.
  • #1
Riemannliness
18
0
(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.
 
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  • #2
Take an orthogonal set of vectors. Add the zero vector to it. What happens?
 
  • #3
Oh snap! Good one.
 
  • #4
Does the book's definition of orthogonal sets allow the 0 vector to be a member?
 
  • #5
Yes, the book takes the stance that the zero vector is orthogonal to every vector.
 
  • #7
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?
 
  • #8
Yes.
 
  • #9
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.
 

1. What is the difference between orthogonal and orthonormal sets?

Orthogonal sets are sets of vectors that are mutually perpendicular, meaning they form right angles with each other. Orthonormal sets are a special type of orthogonal set where all the vectors have a length of 1, making them not only perpendicular but also unit vectors.

2. How do you determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if none of the vectors in the set can be written as a linear combination of the other vectors. In other words, no vector in the set is redundant and all of them are necessary to span the vector space.

3. Can an orthogonal set also be linearly independent?

Yes, an orthogonal set can also be linearly independent. In fact, all orthonormal sets are also linearly independent, as the unit vectors in an orthonormal set cannot be written as a linear combination of each other.

4. What is the importance of linear independence in linear algebra?

Linear independence is important in linear algebra because it allows us to easily solve systems of linear equations and determine unique solutions. It also helps us understand the structure and properties of vector spaces.

5. How can you check if a set of vectors is orthonormal?

To check if a set of vectors is orthonormal, you can calculate the dot product of each pair of vectors. If the dot product is 0, the vectors are orthogonal. Then, you can check if the length of each vector is 1 to determine if they are also unit vectors, making the set orthonormal.

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