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 all projection terms are zero.

Step-by-step solution

Understand the Gram-Schmidt Process

The Gram-Schmidt process is used to convert a set of linearly independent vectors into an orthogonal (or orthonormal) set. Given a basis \(\{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \}\), each new vector \(\mathbf{f}_k\) is computed by removing the projection of \(\mathbf{v}_k\) onto the preceding vectors \(\mathbf{f}_1, \ldots, \mathbf{f}_{k-1}\).

Define the Projection Formula

The projection of a vector \(\mathbf{v}_k\) onto another vector \(\mathbf{f}_j\) is given by \(\text{Proj}_{\mathbf{f}_j}(\mathbf{v}_k) = \frac{\langle \mathbf{v}_k, \mathbf{f}_j \rangle}{\langle \mathbf{f}_j, \mathbf{f}_j \rangle} \mathbf{f}_j\). In the Gram-Schmidt process, the orthogonal component is calculated as \(\mathbf{f}_k = \mathbf{v}_k - \sum_{j=1}^{k-1} \text{Proj}_{\mathbf{f}_j}(\mathbf{v}_k)\).

Apply to the Orthogonal Basis

Since the given basis \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\) is already orthogonal, the projections \(\text{Proj}_{\mathbf{f}_j}(\mathbf{v}_k)\) where \(j < k\) have zero contributions. This is because \(\langle \mathbf{v}_k, \mathbf{v}_j \rangle = 0\) for \(j eq k\) in an orthogonal set.

Simplify Using Orthogonality

For each \(k\), since all projections are zero, the formula simplifies to \(\mathbf{f}_k = \mathbf{v}_k\). There are no contributions from other vectors in the calculation, directly resulting in \(\mathbf{f}_k = \mathbf{v}_k\).

Conclusion

Since \(\mathbf{f}_k = \mathbf{v}_k\) for all \(k = 1, 2, \ldots, n\), the Gram-Schmidt process reproduces the same orthogonal basis when applied to an already orthogonal set of vectors.

Key concepts

Orthogonal Basis

An orthogonal basis is a set of vectors that are not only linearly independent but also at right angles to one another. When vectors are orthogonal, the dot product between any two different vectors equals zero. This property simplifies many calculations in linear algebra and geometry. Orthogonal bases are crucial in various fields, from computer graphics to quantum mechanics, because they make mathematical transformations and equations more manageable.

• Orthogonality: For two vectors to be orthogonal, their dot product must be zero: \(\langle \mathbf{v}_i, \mathbf{v}_j \rangle = 0\) for all \(i eq j\).
• Simplicity: With orthogonal vectors, projections and coordinate transformations become straightforward.
• Efficiency: Reduces computational complexity in algorithms, especially in machine learning and data analysis.

The Gram-Schmidt process capitalizes on this property by starting with any linearly independent set of vectors and systematically turning them into an orthogonal basis. However, if you begin with an orthogonal set, the process reveals that there's nothing left to be done: every original vector in the basis stays the same, because no projections alter them.

Vector Projection

The vector projection is an operation that projects one vector onto another. Imagine shining a light along one vector onto another; the shadow cast represents the projection.

In mathematics, the projection of a vector \(\mathbf{v}_k\) onto a vector \(\mathbf{f}_j\) is described by the formula:\[\text{Proj}_{\mathbf{f}_j}(\mathbf{v}_k) = \frac{\langle \mathbf{v}_k, \mathbf{f}_j \rangle}{\langle \mathbf{f}_j, \mathbf{f}_j \rangle} \mathbf{f}_j\] This formula calculates how much of \(\mathbf{v}_k\) lies in the direction of \(\mathbf{f}_j\). In the context of the Gram-Schmidt process, projecting \(\mathbf{v}_k\) onto each of the preceding vectors \(\mathbf{f}_1, \ldots, \mathbf{f}_{k-1}\) helps to subtract components in those directions, leaving the new vector orthogonal to all previous vectors.

But, as seen in an already orthogonal set, the projections naturally zero out due to the perpendicularity condition, leading to simplifications in calculations.

Linear Independence

Linear independence is a fundamental concept when discussing vector spaces and bases. A set of vectors is said to be linearly independent if no vector in the set can be written as a combination of the others. In simpler terms, each vector brings a new dimension or direction to the space.

• Definition: Vectors \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\) are linearly independent if the equation \(c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \ldots + c_n\mathbf{v}_n = \mathbf{0}\) has only the trivial solution \(c_1 = c_2 = \ldots = c_n = 0\).
• Importance: Ensures all directions in a space are captured without redundancy.
• Application: Key to understanding vector spaces, dimension and solving systems of linear equations.

When you apply the Gram-Schmidt process, you start with a linearly independent set to successfully build an orthogonal basis. This guarantees that each vector you derive adds a new direction or angle and helps avoid redundancy in the basis.