Orthogonality: Projections, Norms, and Gram–Schmidt

Abstract

Orthogonality is a central concept in linear algebra, pivotal for understanding vector spaces and their geometric interpretations. This article explores the definitions and implications of orthogonality, focusing on inner products, orthogonal projections, and the Gram–Schmidt process. By examining these elements, we highlight their interconnections and applications in simplifying complex problems in various fields.

Background

In linear algebra, the inner product is a fundamental operation that allows us to define angles and lengths in vector spaces. For two vectors (\mathbf{u}) and (\mathbf{v}) in (\mathbb{R}^n), the inner product is defined as:

$\langle \mathbf{u}, \mathbf{v} \rangle = \sum_{i=1}^{n} u_i v_i$

This operation provides a measure of similarity between vectors, where an inner product of zero indicates that the vectors are orthogonal, or perpendicular to each other ^{[inner-product]}. Orthogonality is crucial for various applications, including projections and the construction of orthogonal bases.

Orthogonal sets of vectors are collections where each pair of distinct vectors is orthogonal. Formally, a set ({\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n}) is orthogonal if:

$\langle \mathbf{v}_i, \mathbf{v}_j \rangle = 0 \text{ for all } i \neq j$

Such sets simplify the representation of vector spaces and are instrumental in constructing orthogonal bases, which facilitate easier computations in projections and linear combinations ^{[orthogonal-sets]}.

Key results

Inner Products and Orthogonality

The inner product not only defines orthogonality but also underpins the concept of vector length, or norm. The length of a vector (\mathbf{v}) in (\mathbb{R}^n) is given by:

$||\mathbf{v}|| = \sqrt{\sum_{i=1}^{n} v_i^2}$

This norm quantifies how far the vector extends from the origin, and is essential for normalizing vectors to create unit vectors, which have a length of one ^{[length-of-vectors]}, ^{[unit-vector]}.

Orthogonal Projections

Orthogonal projection is a method of projecting a vector onto a subspace, yielding the closest point in that subspace to the original vector. The orthogonal projection of a vector (\mathbf{y}) onto a subspace spanned by an orthogonal set of vectors (\mathbf{u}_1, \mathbf{u}_2, \ldots) is expressed as:

$\tilde{\mathbf{y}} = \frac{\langle \mathbf{y}, \mathbf{u}_1 \rangle}{\langle \mathbf{u}_1, \mathbf{u}_1 \rangle} \mathbf{u}_1 + \frac{\langle \mathbf{y}, \mathbf{u}_2 \rangle}{\langle \mathbf{u}_2, \mathbf{u}_2 \rangle} \mathbf{u}_2 + \ldots$

The distance from the vector (\mathbf{y}) to the subspace is given by (||\mathbf{y} - \tilde{\mathbf{y}}||) ^{[orthogonal-projection]}. This operation is vital in applications such as least squares fitting, where minimizing the error between observed data and a model is essential.

Gram–Schmidt Process

The Gram–Schmidt process is a method for orthogonalizing a set of vectors in an inner product space. Given a linearly independent set of vectors, this process generates an orthogonal set of vectors that span the same subspace. The steps involve taking each vector and subtracting the projections of the previously computed orthogonal vectors, ensuring that the resulting vectors are mutually orthogonal.

Worked examples

Example 1: Orthogonal Projection

Consider a vector (\mathbf{y} = \begin{pmatrix} 3 \ 4 \end{pmatrix}) and an orthogonal set of vectors (\mathbf{u}_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix}) and (\mathbf{u}_2 = \begin{pmatrix} 0 \ 1 \end{pmatrix}). The orthogonal projection of (\mathbf{y}) onto the subspace spanned by (\mathbf{u}_1) and (\mathbf{u}_2) can be computed as follows:

1. Project onto (\mathbf{u}_1): $\tilde{\mathbf{y}}_1 = \frac{\langle \mathbf{y}, \mathbf{u}_1 \rangle}{\langle \mathbf{u}_1, \mathbf{u}_1 \rangle} \mathbf{u}_1 = \frac{3}{1} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 3 \\ 0 \end{pmatrix}$

2. Project onto (\mathbf{u}_2): $\tilde{\mathbf{y}}_2 = \frac{\langle \mathbf{y}, \mathbf{u}_2 \rangle}{\langle \mathbf{u}_2, \mathbf{u}_2 \rangle} \mathbf{u}_2 = \frac{4}{1} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 4 \end{pmatrix}$

Thus, the orthogonal projection of (\mathbf{y}) onto the subspace is:

$\tilde{\mathbf{y}} = \tilde{\mathbf{y}}_1 + \tilde{\mathbf{y}}_2 = \begin{pmatrix} 3 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 4 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$

Example 2: Gram–Schmidt Process

Given two linearly independent vectors (\mathbf{v}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix}) and (\mathbf{v}_2 = \begin{pmatrix} 1 \ 0 \end{pmatrix}), we can apply the Gram–Schmidt process:

1. Set (\mathbf{u}_1 = \mathbf{v}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix}).

2. Compute (\mathbf{u}_2): $\mathbf{u}_2 = \mathbf{v}_2 - \frac{\langle \mathbf{v}_2, \mathbf{u}_1 \rangle}{\langle \mathbf{u}_1, \mathbf{u}_1 \rangle} \mathbf{u}_1$ First, calculate the inner products: $\langle \mathbf{v}_2, \mathbf{u}_1 \rangle = 1 \cdot 1 + 0 \cdot 1 = 1$ $\langle \mathbf{u}_1, \mathbf{u}_1 \rangle = 1^2 + 1^2 = 2$ Thus, $\mathbf{u}_2 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} - \frac{1}{2} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 - 0.5 \\ 0 - 0.5 \end{pmatrix} = \begin{pmatrix} 0.5 \\ -0.5 \end{pmatrix}$

The resulting orthogonal set is ({\mathbf{u}_1, \mathbf{u}_2}).

References

• ^{[inner-product]}
• ^{[orthogonal-projection]}
• ^{[orthogonal-sets]}
• ^{[length-of-vectors]}
• ^{[unit-vector]}

AI disclosure

This article was generated with the assistance of AI, based on personal class notes and structured to provide a clear and concise overview of the topic of orthogonality in linear algebra.