Question

Using the Gram-Schmidt procedure: (a) construct an orthonormal set of vectors from the following: (b) find an orthonormal basis, within a four-dimensional Euclidean space, for the subspace spanned by the three vectors \(\left(\begin{array}{llll}1 & 2 & 0 & 0\end{array}\right)^{\mathrm{T}},(3 \quad-1 \quad 2 \quad 0)^{\mathrm{T}}\) and \(\left.\begin{array}{llll}0 & 0 & 2 & 1\end{array}\right)^{\mathrm{T}}\).

Short answer

The orthonormal basis consists of vectors \(\mathbf{e}_1\), \(\mathbf{e}_2\), and \(\mathbf{e}_3\).

Step-by-step solution

- Identify Initial Vectors

Given vectors are \(\mathbf{v}_1 = \left(\begin{array}{c}1 \2 \0 \0\end{array}\right)^{\mathrm{T}}\,\mathbf{v}_2 = \left(\begin{array}{c}3 \-1 \2 \0\end{array}\right)^{\mathrm{T}}\,\mathbf{v}_3 = \left(\begin{array}{c}0 \0 \2 \1\end{array}\right)^{\mathrm{T}}\).

- Gram-Schmidt Process: First Vector

Set \(\mathbf{u}_1 = \mathbf{v}_1\). Normalize \(\mathbf{u}_1\) to get the first orthonormal vector: \(\mathbf{e}_1 = \frac{\mathbf{u}_1}{\|\mathbf{u}_1\|} = \frac{1}{\sqrt{5}} \left(\begin{array}{c}1 \ 2 \ 0 \ 0\end{array}\right)\).

- Gram-Schmidt Process: Second Vector

Compute \(\mathbf{u}_2\) by subtracting the projection of \(\mathbf{v}_2\) on \(\mathbf{e}_1\) from \(\mathbf{v}_2\): \ \(\mathbf{u}_2=\mathbf{v}_2 - (\mathbf{v}_2 \cdot \mathbf{e}_1) \mathbf{e}_1\). Calculate the projection: \ \( \mathbf{v}_2 \cdot \mathbf{e}_1 = \left(\begin{array}{c}3 \ -1 \ 2 \ 0 \end{array}\right) \cdot \frac{1}{\sqrt{5}} \left(\begin{array}{c}1 \ 2 \ 0 \ 0 \end{array}\right) = \frac{1}{\sqrt{5}} (3 - 2) = \frac{1}{\sqrt{5}} \). Thus, \ \( \mathbf{u}_2 = \left(\begin{array}{c}3 \ -1 \ 2 \ 0 \end{array}\right) - \frac{1}{\sqrt{5}} \left(\begin{array}{c}1 \ 2 \ 0 \ 0 \end{array}\right) = \left(\begin{array}{c}\frac{14}{5} \ -\frac{7}{5} \ 2 \ 0 \end{array}\right)\). Normalize \(\mathbf{u}_2\) to obtain \(\mathbf{e}_2 = \frac{\mathbf{u}_2}{\|\mathbf{u}_2\|}\).

- Gram-Schmidt Process: Third Vector

Compute \(\mathbf{u}_3\) by subtracting the projections of \(\mathbf{v}_3\) on both \(\mathbf{e}_1\) and \(\mathbf{e}_2\) from \(\mathbf{v}_3\): \ \(\mathbf{u}_3 = \mathbf{v}_3 - (\mathbf{v}_3 \cdot \mathbf{e}_1) \mathbf{e}_1 - (\mathbf{v}_3 \cdot \mathbf{e}_2) \mathbf{e}_2\). \ First, calculate \(\mathbf{v}_3\) projections: \ \(\mathbf{v}_3 \cdot \mathbf{e}_1 = 0\) and \(\mathbf{v}_3 \cdot \mathbf{e}_2\). Solve: \ \(\mathbf{u}_3 = \mathbf{v}_3 - (\mathbf{v}_3 \cdot \mathbf{e}_2) \mathbf{e}_2 = \left(\begin{array}{c}0 \ 0 \ 2 \ 1\end{array}\right)\). Normalize \(\mathbf{u}_3\) to obtain \(\mathbf{e}_3\).

Key concepts

Orthonormal Basis

An orthonormal basis is a set of vectors that are both orthogonal (i.e., the dot product of any two different vectors is zero) and normalized (i.e., the length or norm of each vector is one) in a given vector space. This concept is pivotal in various fields like quantum mechanics, computer graphics, and machine learning.
When we create an orthonormal basis using the Gram-Schmidt process, we start with a set of linearly independent vectors. These vectors are then orthogonalized and normalized step by step.
For example, consider the vectors provided in the exercise. The process starts with the first vector, normalizes it, and then systematically adjusts each subsequent vector to be orthogonal to all previously processed vectors, followed by normalization.
The result is a new set of vectors that span the same subspace as the original vectors but have the special property of being orthonormal. This enables simpler geometric interpretations and computations, as the transformation properties of orthonormal sets are easier to manage.

Euclidean Space

Euclidean space is a concept from geometry that generalizes the notion of 2D and 3D spaces to any number of dimensions. It encompasses the familiar idea of distance and angle, which we use extensively in calculations and drawings.
Formally, a Euclidean space of dimension is the set of all possible n-tuples of real numbers, denoted as . This space is equipped with a standard dot product that helps in defining notions like the length of a vector and the angle between vectors.
In a four-dimensional Euclidean space, we deal with vectors that have four components. The exercise provided is situated in such a space, with vectors having the form . Operations like the dot product and vector normalization are conducted similarly to how they're done in lower dimensions, albeit with more components.
Understanding Euclidean space is essential when performing vector operations, as it provides the framework in which these operations are defined. It also allows for generalization to higher dimensions, which is a common requirement in fields like data science and physics.

Vector Normalization

Vector normalization is the process of converting a vector to a unit vector, which has a length or norm of one. This is achieved by dividing the vector by its magnitude.
The magnitude (or norm) of a vector , denoted as , is computed as . To normalize a vector , you transform it into .
In the context of the Gram-Schmidt process, normalization is crucial after orthogonalizing each vector. It ensures that the final orthonormal basis consists of unit vectors, which simplifies many mathematical operations and helps in maintaining numerical stability.
For instance, consider the provided vectors in the exercise. The first step involved normalizing to create the unit vector . This step is repeated for every new vector obtained via the orthogonalization process.
Vector normalization is a foundational concept in linear algebra and vector calculus, playing a critical role in algorithms for numerical analysis, machine learning, and computer vision.