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vooor
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can someone help me to solve this problem?
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Finding orthonormal basis for the intersection of the subspaces
- Thread starter vooor
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In summary, an orthonormal basis is a set of mutually perpendicular and unit vectors that span a space. It is important to find an orthonormal basis for the intersection of subspaces because it simplifies calculations and helps us understand the relationship between the subspaces. The Gram-Schmidt process is used to find an orthonormal basis for the intersection, but there can be multiple possible bases. However, there are special cases where the intersection may not have an orthonormal basis. This can occur if the subspaces are not orthogonal or if one is contained within the other. In these cases, the intersection may only have a spanning set, but not a basis.
can someone help me to solve this problem?
I couldn't even approach
- #2
Char. Limit
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Well, you know what a basis is. Because you're looking for one for the intersection of the two subspaces, you know that whatever basis you find has to fit BOTH. So how would you find an orthonormal basis for one of them?
Related to Finding orthonormal basis for the intersection of the subspaces
What is an orthonormal basis?
An orthonormal basis is a set of vectors that are mutually perpendicular (orthogonal) and have a length of 1 (unit vectors). In other words, they are a set of linearly independent vectors that span a space.
Why is it important to find an orthonormal basis for the intersection of subspaces?
Finding an orthonormal basis for the intersection of subspaces allows us to simplify calculations and make them more efficient. It also helps us understand the structure of the intersection and its relationship to the original subspaces.
How do you find an orthonormal basis for the intersection of subspaces?
To find an orthonormal basis for the intersection of subspaces, we use the Gram-Schmidt process. This involves taking a basis for one subspace and orthogonalizing it with respect to the other subspace. The resulting vectors will form an orthonormal basis for the intersection.
Can the intersection of subspaces have more than one orthonormal basis?
Yes, the intersection of subspaces can have multiple orthonormal bases. This is because the Gram-Schmidt process can be applied in different ways, resulting in different sets of orthonormal vectors.
Are there any special cases when finding an orthonormal basis for the intersection of subspaces?
Yes, there are special cases where the intersection of subspaces may not have an orthonormal basis. This can happen if the subspaces are not orthogonal to each other or if one subspace is contained within the other. In these cases, the intersection may only have a spanning set, but not a basis.
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