Is a set of orthogonal basis vectors for a subspace unique?

In summary, the conversation discussed the uniqueness of a set of orthogonal basis vectors for a subspace. The question was whether a subspace had a unique set of basis vectors, and examples were given to show that there are multiple possible sets of orthogonal basis vectors. The conversation also touched on the possibility of finding different sets of basis vectors by rotating them.
  • #1
DryRun
Gold Member
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Homework Statement
Is a set of orthogonal basis vectors for a subspace unique?

The attempt at a solution
I don't know what this means. Can someone please explain?
I managed to find the orthogonal basis vectors and afterwards determining the orthonormal basis vectors, but I'm not sure what the question is asking for exactly?
 
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  • #2
Unique means only one. So, for example, in R2, (1,0) and (0,1) are an orthogonal basis. Can you find a different pair giving an orthogonal bases for R2?
 
  • #3
LCKurtz said:
Unique means only one. So, for example, in R2, (1,0) and (0,1) are an orthogonal basis. Can you find a different pair giving an orthogonal bases for R2?
Thanks for your time, LCKurtz.
To answer your question... Yes, (0,-1) and (-1,0) as well as the following pair (0,0) and (-1,0) and others, like (-1,0) and (0,2), etc.
But what is your point? I don't see how that answers my original question.
 
  • #4
sharks said:
Thanks for your time, LCKurtz.
To answer your question... Yes, (0,-1) and (-1,0) as well as the following pair (0,0) and (-1,0) and others, like (-1,0) and (0,2), etc.
But what is your point? I don't see how that answers my original question.

You don't?? Your question was whether a subspace had a unique (meaning only one) set of basis vectors. You have just mentioned several. And you could find entirely different ones by rotating them too.
 
  • #5
OK, i missed that point, as my mind was focused more on the actual orthogonal basis vectors, which is a 3x3 matrix, which i assume would not be unique either. Thanks!
 

1. What is a set of orthogonal basis vectors for a subspace?

A set of orthogonal basis vectors for a subspace is a set of vectors that are both linearly independent and orthogonal to each other. This means that they span the entire subspace and no vector in the set can be written as a linear combination of the other vectors.

2. Why is it important for a set of orthogonal basis vectors to be unique?

Having a unique set of orthogonal basis vectors is important because it allows for a clear and concise representation of the subspace. It also ensures that any vector in the subspace can be uniquely represented by a linear combination of the basis vectors.

3. How is the uniqueness of a set of orthogonal basis vectors proven?

The uniqueness of a set of orthogonal basis vectors can be proven by using the Gram-Schmidt process. This process takes a set of linearly independent vectors and transforms them into a set of orthogonal vectors, thus ensuring their uniqueness.

4. Can a set of orthogonal basis vectors for a subspace be non-orthogonal?

No, a set of orthogonal basis vectors for a subspace must be orthogonal to each other in order to be considered a valid basis. If the vectors are not orthogonal, they can be transformed into an orthogonal set using the Gram-Schmidt process.

5. How does the concept of orthogonal basis vectors relate to linear independence?

A set of orthogonal basis vectors is always linearly independent. This is because if the vectors were not linearly independent, they would not be able to span the entire subspace and would not be considered a valid basis. Additionally, the Gram-Schmidt process ensures that the resulting orthogonal vectors are linearly independent.

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