Question

If the Gram-Schmidt process is used on an orthogonal basis \(\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\\}\) of \(V,\) show that \(\mathbf{f}_{k}=\mathbf{v}_{k}\) holds for each \(k=1,2, \ldots, n .\) That is, show that the algorithm reproduces the same basis.

Short answer

The Gram-Schmidt process reproduces the same orthogonal basis because projections are zero.

Step-by-step solution

Understanding Gram-Schmidt Process

The Gram-Schmidt process is a method for converting a set of vectors into an orthogonal (or orthonormal) basis. Given vectors \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\), the process generates vectors \(\{\mathbf{f}_1, \mathbf{f}_2, \ldots, \mathbf{f}_n\}\) such that these vectors are orthogonal and span the same subspace.

Define the Orthogonal Basis

Given that \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\) is already an orthogonal basis of \(V\), it means that each vector is orthogonal to the others, i.e., \(\mathbf{v}_i \cdot \mathbf{v}_j = 0\) for \(i eq j\). Each vector is already in the form required by Gram-Schmidt.

Apply Gram-Schmidt to \(\mathbf{v}_1\)

For the first vector, the Gram-Schmidt process sets \(\mathbf{f}_1 = \mathbf{v}_1\). Since \(\mathbf{v}_1\) is already part of an orthogonal basis, no alteration occurs.

Apply Gram-Schmidt to \(\mathbf{v}_2\)

For the second vector, \(\mathbf{f}_2 = \mathbf{v}_2 - \text{proj}_{\mathbf{f}_1}(\mathbf{v}_2)\). Since the basis is orthogonal, the projection of \(\mathbf{v}_2\) onto \(\mathbf{f}_1\) is zero. Therefore, \(\mathbf{f}_2 = \mathbf{v}_2\).

Generalize for \(\mathbf{v}_k\)

Continuing this process, for each \(k\), \(\mathbf{f}_k = \mathbf{v}_k - \sum_{j=1}^{k-1} \text{proj}_{\mathbf{f}_j}(\mathbf{v}_k)\). Again, since the basis is orthogonal, projections \(\text{proj}_{\mathbf{f}_j}(\mathbf{v}_k)\) are all zero, so \(\mathbf{f}_k = \mathbf{v}_k\).

Conclusion

Since every vector in the original orthogonal basis \(\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}\) remains unchanged after applying the Gram-Schmidt process, \(\mathbf{f}_k = \mathbf{v}_k\) for all \(k=1,2,\ldots,n\), thus reproducing the same basis.

Key concepts

Orthogonal Basis

An orthogonal basis is a set of vectors such that each pair of different vectors is orthogonal. This means that the dot product between any two distinct vectors in the set is zero. This property is essential because it implies that these vectors are linearly independent.
Orthogonal bases are particularly useful in simplifying many mathematical computations. For example, when you work with matrices or linear transformations, having an orthogonal basis makes it easier to perform operations like projections. Moreover, if you add the condition that each vector must have a length of one, the basis becomes orthonormal.
To summarize, the orthogonal nature guarantees no overlap or redundancy among the vectors, ensuring each vector contributes a unique direction in space.

Orthonormal Basis

An orthonormal basis takes the concept of an orthogonal basis and takes it a step further. In an orthonormal basis, not only are the vectors orthogonal, they are also normalized, meaning each vector has a magnitude of one. This is accomplished by dividing each orthogonal vector by its magnitude.
Orthonormal bases are especially beneficial because they make calculations involving angles and lengths straightforward. For instance, the simple dot product of two orthonormal vectors immediately yields the cosine of the angle between them. Furthermore, the inverse of an orthonormal matrix is simply its transpose, making matrix operations much more efficient.
In short, an orthonormal basis is both intuitive and practical, providing a convenient framework for mathematical and computational methodologies.

Vector Projection

Vector projection is a way of projecting one vector onto another vector. This involves calculating how much of one vector goes in the direction of another. The result is a new vector lying on the line defined by the second vector.
To find the projection of a vector \(\mathbf{v}\) onto vector \(\mathbf{u}\), we use the formula: \[ \text{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} \] This formula gives us the component of \(\mathbf{v}\) that is in the direction of \(\mathbf{u}\). This concept is pivotal in the Gram-Schmidt process, as it allows the creation of orthogonal vectors step by step.
Understanding vector projections helps in breaking down complex vector interactions into simpler, orthogonal components, facilitating easier manipulation and analysis of vector sets.