# Matrix which reverses Gram-Schmidt - Linear Algebra

In summary, the conversation discusses using the Gram-Schmidt method to find the orthogonal q vectors in terms of input vectors, rewriting equations to use q vectors, and finding a matrix to represent the transformation. However, due to linear independence, the q vectors cannot be written in terms of previous vectors.
Homework Statement
Let a1, a2, a3 be linearly independent vectors in R 3 , and let q1, q2, q3 be the vectors obtained from a1, a2, a3 by the Gram-Schmidt algorithm.

Define the linear transformation T : R 3 → R 3 by T(q1) = a1, T(q2) = a2 and T(q3) = a3.

For the choice of basis {q1, q2, q3} both for the input and output spaces, find the matrix MT which represents the linear transformation T for this choice of basis.
Relevant Equations
projection equation used in Gram-Schmidt: p=((a^t)(b))/((a^t)(a)) (a)
My idea was to write out the formulas for the orthogonal q vectors in terms of the input vectors using the basics of gram-schmidt. Then, I would rewrite those equations suhc that the a vectors were written in terms of the q vectors. And then, try to find some matrix which would capture the needed transformation. However, the q vectors must be written in terms of multiple a vectors, as well as other q vectors. Additionally, the need to make all of the orthogonal vecotrs of unit length makes it even more complicated. There must be another way.

If a1-3 are linearly independent vectors, shouldn't q1 = a1, q2 = a2, q3 = a3?

Because Gram-Schmidt method is about substracting the part of your vector that's represented in previous vectors. But there are no such parts since they are linearly independent.

## 1. What is the Matrix which reverses Gram-Schmidt in Linear Algebra?

The Matrix which reverses Gram-Schmidt in Linear Algebra is a matrix that undoes the process of Gram-Schmidt orthogonalization, which is a method used to find an orthogonal basis for a given set of vectors. This matrix is also known as the inverse of the Gram-Schmidt matrix.

## 2. Why is the Matrix which reverses Gram-Schmidt important in Linear Algebra?

The Matrix which reverses Gram-Schmidt is important in Linear Algebra because it allows us to undo the process of Gram-Schmidt orthogonalization and retrieve the original set of vectors. This is useful in many applications, such as in solving systems of linear equations and in finding the best fit for a set of data points.

## 3. How is the Matrix which reverses Gram-Schmidt calculated?

The Matrix which reverses Gram-Schmidt can be calculated by taking the transpose of the matrix formed by the orthogonal basis vectors found through Gram-Schmidt orthogonalization. This is because the transpose of a matrix is the inverse of the matrix if it is orthogonal.

## 4. Can the Matrix which reverses Gram-Schmidt be used for any set of vectors?

No, the Matrix which reverses Gram-Schmidt can only be used for sets of linearly independent vectors. This is because the Gram-Schmidt process can only be applied to linearly independent vectors, and the resulting orthogonal basis is unique. If the original set of vectors is not linearly independent, then the process cannot be reversed.

## 5. Are there any limitations to using the Matrix which reverses Gram-Schmidt?

One limitation of using the Matrix which reverses Gram-Schmidt is that it can only be used for finite-dimensional vector spaces. This means that it cannot be applied to infinite-dimensional vector spaces, such as function spaces. Additionally, the matrix may be computationally expensive to calculate for large sets of vectors.

• Calculus and Beyond Homework Help
Replies
8
Views
960
• Calculus and Beyond Homework Help
Replies
16
Views
2K
• Linear and Abstract Algebra
Replies
14
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
3K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Linear and Abstract Algebra
Replies
19
Views
731
• Calculus and Beyond Homework Help
Replies
8
Views
2K