Gram-Schmidt Process: Example 3 in PDF

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In summary, the authors attempted to solve for the second orthonormal basis vector using the Gram-Schmidt process but were unsuccessful. They found that if they normalized the vector using this process, they ended up with a different vector than what was given.
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Homework Statement


Example 3 in this pdf: http://karin.fq.uh.cu/qct/Tema_04/0...es propios de un sistema/Diagonalization.pdf"

Homework Equations


Gram-schmidt process:
v2 perp = v2 - (u1*v2)u1


The Attempt at a Solution


I don't understand how they got the second orthonormal basis vector. Using the equation above, with u1=1/sqrt(2)[-1 1 0] and v2=[-1 0 1] (these vectors are vertical), I don't get what they got as the second basis vector. Am I doing the Gram-Schmidt process the wrong way?
 
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  • #2
Any non-zero multiple of an eigenvector is also an eigenvector. The results in the examples have been normalized. Did you normalize yours?
 
  • #3
The examples state that the normalization has been done through Gram-Schmidt but I don't get the same results when I try to normalize them with Gram-Schmidt. Is there another way of normalizing?
 
  • #4
Gram-Schmidt doesn't normalize; it orthogonalizes.
 
  • #5
Okay, so if I normalize [-1 1 0], I get 1/sqrt(2)[-1 1 0] which is what the examples indicate. However, if I normalize [-1 0 1], I get 1/sqrt(2)[-1 0 1], which doesn't match the example. I don't get how they got 1/sqrt(6)[-1 -1 2] from [-1 0 1].
 
  • #6
Try normalizing [-1 1 2].
 
  • #7
Where did you get [-1 1 2]? Why can't 1/sqrt(2)[-1 0 1] be a vector in the orthonormal basis?
 
  • #8
Let's back up a second. Did you calculate the second (unnormalized) vector using the Gram-Schmidt process yet? It's this vector that you want to normalize, not the given vector [-1 0 1].

Also, I meant [-1 -1 2] earlier, not [-1 1 2]. I accidentally dropped a sign. You mentioned [-1 -1 2] in your earlier post.
 
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  • #9
Using the Gram-Schmidt process, I get [-1/2 -1/2 1] as the unnormalized second vector. So am I allowed to multiply this by 2 and normalize that to get the second vector in the orthonormal basis?
 
  • #10
Yup, though you don't need to multiply by 2 first. You can normalize it as is.
 
  • #11
Okay. Thank you so much for working through this with me!
 

1. What is the Gram-Schmidt process?

The Gram-Schmidt process is a mathematical method used to orthogonalize a set of vectors in a vector space. It is commonly used in linear algebra to transform a set of linearly independent vectors into a set of orthogonal vectors.

2. How does the Gram-Schmidt process work?

The process involves taking a set of vectors and creating a new set of orthogonal vectors by subtracting the projections of the original vectors onto the already created orthogonal vectors. This process is repeated until all the vectors in the original set are orthogonal.

3. Why is the Gram-Schmidt process useful?

The Gram-Schmidt process is useful because it allows for easier computation and analysis of vectors. By creating a set of orthogonal vectors, it simplifies many calculations and allows for a clearer understanding of the relationships between vectors in a vector space.

4. Can you provide an example of the Gram-Schmidt process?

Yes, in the PDF example provided, there is a step-by-step demonstration of the Gram-Schmidt process being applied to a set of vectors. The example shows how the original set of vectors is transformed into a set of orthogonal vectors through repeated calculations and projections.

5. In what fields is the Gram-Schmidt process commonly used?

The Gram-Schmidt process is commonly used in fields such as linear algebra, mathematics, physics, and engineering. It is also utilized in computer graphics and signal processing to simplify vector calculations and improve the accuracy of results.

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