Gram-Schmidt Orthonormalization: Modifying Vectors & Spanning Space

In summary, Gram-Schmidt Orthonormalization is a mathematical process used to modify a set of linearly independent vectors into a set of orthonormal vectors. Its main purpose is to create a set of orthonormal vectors that can be used as a basis for a vector space. This process involves finding the orthogonal projections of each vector onto the span of the previous vectors, subtracting them from the original vectors, and normalizing them to have a length of 1. It can be used to span a space by applying the Gram-Schmidt process to a set of linearly independent vectors. However, it can only be applied to linearly independent sets, as it will result in a division by 0 if the set is
  • #1
dntmn
2
0
I have a question about Gram Schmidt Orthonormalization. I know you can orthogonalize then normalize.

My question is can you (multipy/divide) the given vectors by a scalar to give new vectors, then orthogonalize(by taking projection), then normalize the result?

would the result still span the space of the orgional vectors?
 
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  • #2
Of course, just imagine the scalar hanging out front the whole time, then it goes away when you normalize at the end.
 
  • #3
Thank you!
 
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1. What is Gram-Schmidt Orthonormalization?

Gram-Schmidt Orthonormalization is a mathematical process used to modify a set of linearly independent vectors into a set of orthonormal vectors. This process involves finding the orthogonal projections of each vector onto the span of the previous vectors and then normalizing them to have a length of 1.

2. What is the purpose of Gram-Schmidt Orthonormalization?

The main purpose of Gram-Schmidt Orthonormalization is to create a set of orthonormal vectors that can be used as a basis for a vector space. This is useful in many applications, such as solving systems of linear equations, computing eigenvalues and eigenvectors, and performing transformations in linear algebra.

3. How does Gram-Schmidt Orthonormalization modify vectors?

Gram-Schmidt Orthonormalization modifies vectors by first finding the orthogonal projections of each vector onto the span of the previous vectors. Then, it subtracts these projections from the original vectors to make them orthogonal to each other. Finally, the modified vectors are normalized to have a length of 1.

4. What is the process for using Gram-Schmidt Orthonormalization to span a space?

The process for using Gram-Schmidt Orthonormalization to span a space involves taking a set of linearly independent vectors and applying the Gram-Schmidt process to them. This will result in a set of orthonormal vectors that span the same space as the original vectors. The orthonormal vectors can then be used as a basis for the vector space.

5. Can Gram-Schmidt Orthonormalization be applied to any set of vectors?

No, Gram-Schmidt Orthonormalization can only be applied to sets of linearly independent vectors. If the set of vectors is linearly dependent, the process will result in a division by 0 and cannot be completed. It is important to check for linear independence before applying Gram-Schmidt Orthonormalization.

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