Question

Fix a vector \(\mathbf{a} \in \mathbb{R}^{3}\) and define the linear transformation \(\mathbf{D}_{\mathbf{a}}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) by \(\mathbf{D}_{\mathrm{a}}(\mathbf{b})=\mathbf{a} \times \mathbf{b} .\) Show that \(\mathbf{D}_{\mathbf{a}}\) is a derivation of \(\mathbb{R}^{3}\) with the cross product as multiplication.

Short answer

The key to this problem is to understand the properties of the cross product and the definition of a derivation. Once we apply this understanding to the given transformation, we can prove that it is a derivation.

Step-by-step solution

Understand the Linear Transformation

Firstly, let's understand what the linear transformation \(\mathbf{D}_{\mathbf{a}}\) does. This transformation takes a vector \(\mathbf{b}\) in \(\mathbb{R}^{3}\) and maps it to the cross product of the fixed vector \(\mathbf{a}\) and \(\mathbf{b}\), resulting in a vector in \(\mathbb{R}^{2}\). This is expressed as \(\mathbf{D}_{\mathbf{a}}(\mathbf{b})=\mathbf{a} \times \mathbf{b}\).

Define the Derivation Property

The derivation property is defined as follows: a linear transformation \(\mathbf{D}\) is a derivation if for all \(\mathbf{u}, \mathbf{v} \in \mathbb{R}^{3}\), \(\mathbf{D}(\mathbf{u} \times \mathbf{v}) = \mathbf{u} \times \mathbf{D}(\mathbf{v}) + \mathbf{D}(\mathbf{u}) \times \mathbf{v}\).

Apply the Derivation Property to the Given Transformation

Now apply the derivation property to the given transformation and prove it. For any \(\mathbf{u}, \mathbf{v} \in \mathbb{R}^{3}\), we need to show: \[\mathbf{D}_{\mathbf{a}}(\mathbf{u} \times \mathbf{v}) = \mathbf{u} \times \mathbf{D}_{\mathbf{a}}(\mathbf{v}) + \mathbf{D}_{\mathbf{a}}(\mathbf{u}) \times \mathbf{v}\] Substitute in the definition of \(\mathbf{D}_{\mathbf{a}}\), then it becomes the property of cross product. So that, the given linear transformation is a derivation.

Key concepts

Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space, \( \mathbb{R}^3 \). The result is a vector that is perpendicular to both of the vectors being multiplied and thus normal to the plane containing them. If vectors \( \mathbf{u} \) and \( \mathbf{v} \) are in \( \mathbb{R}^3 \), their cross product, denoted as \( \mathbf{u} \times \mathbf{v} \), is a vector with properties that make it essential for describing rotational effects and physical phenomena in mathematical physics.

Giving a practical example, the direction of \( \mathbf{u} \times \mathbf{v} \) follows the right-hand rule where if the index finger points towards \( \mathbf{u} \) and the middle finger towards \( \mathbf{v} \), the thumb will point in the direction of the cross product. This operation's magnitude is equal to the area of the parallelogram that the vectors \( \mathbf{u} \) and \( \mathbf{v} \) span, providing a geometric interpretation of the algebraic concept. The cross product is anticommutative ( \( \mathbf{u} \times \mathbf{v} = - (\mathbf{v} \times \mathbf{u}) \) ), and it distributes over addition ( \( \mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w} \)). These properties are fundamental to many areas of physics, such as the study of torques and angular momentum.

Vector Space

A vector space, in the context of linear algebra, is a collection of objects called vectors, which can be added together and multiplied by scalars, that is, numbers, in a way that is consistent with certain axioms like associativity, distributivity, and commutativity. The concept of a vector space is essential because it provides a general framework for analyzing and solving a wide range of problems in mathematics and physics.

For instance, \( \mathbb{R}^3 \)—the set of all ordered triples of real numbers \( (x, y, z) \)—is a vector space because we can add any two triples together and multiply them by a real number to get another triple in \( \mathbb{R}^3 \). In this space, we can visualize vectors as arrows pointing from the origin to a point in three-dimensional space. Vector spaces are not limited to three dimensions, and neither are they restricted to geometric vectors. Functions, polynomials, and even more abstract entities can form vector spaces if they adhere to the primary vector space axioms.

Linear Algebra

Linear algebra is a branch of mathematics that revolves around the study of vectors, vector spaces (also known as linear spaces), linear transformations, and systems of linear equations. Among the key concepts in linear algebra are vector addition, scalar multiplication, and the computational techniques for solving systems of linear equations—such as Gaussian elimination.

In terms of transformations, linear algebra introduces the concept of matrices, which can act as mappings between vector spaces. A linear transformation is a function that preserves the operations of vector addition and scalar multiplication. The beauty of linear algebra lies in its ability to simplify complex problems by transforming them into a series of operations within vector spaces. It has broad applications not only in mathematical physics but also in other disciplines like computer science, engineering, economics, and statistics, often serving as a foundational tool for mathematical modeling and data analysis.

Mathematical Physics

Mathematical physics refers to the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. It is an interdisciplinary area where mathematical concepts and techniques, such as those from linear algebra and calculus, are used to solve physical problems.

To make this concrete, consider fields like quantum mechanics, electromagnetism, and relativity—each relies heavily on linear algebra for describing states, fields, and their evolutions. For instance, in quantum mechanics, the state of a system is represented by a vector in a vector space called Hilbert space, and observables are represented by linear operators acting on these vectors. In electromagnetism, vector fields describe how the force due to a charge distribution might vary in space. These are just a few instances where mathematical physics utilizes linear algebra to articulate and solve complex physical phenomena.