Question

Berechnen Sie die ko- und kontravarianten Komponenten des Ortsvektors \(\mathrm{x}\) des Punktes \(P=(2,4,1)\) bezüglich der Basis \(\mathbf{e}_{1}=(2,1,1)^{T}, \mathbf{e}_{2}=(3,1,1)^{T}\), \(\mathbf{e}_{3}=(4,1,2)^{T}\)

Short answer

Find contravariant components using the inverse basis matrix on point P.

Step-by-step solution

Understand the Coordinate System

We are given a point \(P = (2,4,1)\) in a standard coordinate system. The task is to find the components of this point in a new basis given by \(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\). This involves changing the basis from the standard to the new basis forms given.

Check Linear Independence of the Basis

To ensure that the new basis forms are valid, we must check if \(\mathbf{e}_{1} = (2,1,1)^T\), \(\mathbf{e}_{2} = (3,1,1)^T\), and \(\mathbf{e}_{3} = (4,1,2)^T\) are linearly independent. This is done by calculating the determinant of the matrix formed by these vectors. If it's not zero, they form a suitable basis.

Form the Transformation Matrix

Construct a matrix \(E\) using the vectors \(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\) as columns: \[ E = \begin{pmatrix} 2 & 3 & 4 \ 1 & 1 & 1 \ 1 & 1 & 2 \end{pmatrix} \]

Find the Inverse of the Transformation Matrix

Calculate the inverse of the matrix \(E\), denoted as \(E^{-1}\). This inverse will help in transforming the point \(P\) from the original basis to the new basis.

Calculate Contravariant Components

To find the contravariant components of the point \(P\), calculate \[ x_{contra} = E^{-1} \cdot \begin{pmatrix} 2 \ 4 \ 1 \end{pmatrix} \] This gives us the coordinates in the new basis.

Calculate Covariant Components

Once we know the contravariant components, convert these to covariant components if needed by using these components along with the metric tensor. In Cartesian systems, contravariant and covariant components are typically the same, but in non-Cartesian systems, they may differ depending on the metric tensor.

Key concepts

Coordinate Transformation

In linear algebra, a coordinate transformation is the process of translating the coordinates of a point from one basis to another. Imagine a city map where you use different grid systems to locate places. Similarly, in mathematics, a point like \( P = (2, 4, 1) \) can be expressed in terms of different sets of basis vectors.
In our task, we're transforming the coordinates of a point from the standard basis into a new basis made up of vectors \( \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \). This new basis changes our perspective of the vector without moving the actual point in space. To achieve this transformation, we use a transformation matrix, which is essentially a roadmap telling you how to adjust the coordinates.

Basis Vectors

Basis vectors are the building blocks of any vector space. They span the entire space, which means you can combine them in different ways to express any point in the space.
For example, the standard Cartesian coordinates \( (2, 4, 1) \) are built using the standard basis vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \). A basis set is legitimate if the vectors in it are linearly independent, meaning none of them can be written as a combination of the others.

• Given vectors \( \mathbf{e}_1 = (2, 1, 1)^T \), \( \mathbf{e}_2 = (3, 1, 1)^T \), \( \mathbf{e}_3 = (4, 1, 2)^T \) form a new basis for our vector space.
• Checking them for linear independence means calculating the determinant of the matrix formed by these vectors.

If the determinant is non-zero, these vectors offer a valid basis to describe the space.

Contravariant and Covariant Components

When transforming coordinates, it’s important to distinguish between contravariant and covariant components. Think of them as different ways of measuring with rulers that stretch in opposite directions.

• **Contravariant components**: These are like the default, traditional way you think of coordinates. They scale directly with your basis vectors. To compute them, you need the inverse of the transformation matrix \( E^{-1} \).
• **Covariant components**: These are often related to how measurements would adjust if the geometry of space changed. In simple Cartesian systems, covariant components are usually identical to contravariant ones because of the straightforwardness of the metric tensor. In more complex, non-Cartesian systems, however, they differ as a result of the interaction with the metric tensor.

For the exercise, obtaining the contravariant components involves multiplying \( E^{-1} \) with the point vector \((2, 4, 1)\).

Matrix Inversion

A core part of coordinate transformation is matrix inversion. Simply put, a matrix inverse is like a mathematical undo button. It reverses transformations, allowing you to convert between different coordinate systems. Consider our transformation matrix \( E \) formed by the new basis vectors as columns:
\[ E = \begin{pmatrix} 2 & 3 & 4 \ 1 & 1 & 1 \ 1 & 1 & 2 \end{pmatrix} \]
You find \( E^{-1} \), the inverse of this matrix, to translate the coordinates of \( P \) into the new basis.

• To calculate the inverse of a matrix, ensure that the matrix is square and its determinant is non-zero.
• Once you have \( E^{-1} \), you can use it to compute the contravariant components efficiently.

This step equips you with the ability to not only transform coordinates into a new basis but also explore how those coordinates play out in the broader scheme of spatial dimensions.