Question

Let \(\left\\{\mathbf{e}_{i}\right\\}_{i=1}^{N}\) be a \(g\) -orthonormal basis of \(\mathcal{V}\). Let \(\boldsymbol{\eta}\) be the matrix with elements \(\eta_{i j}\), which is the matrix of \(\mathbf{g}\) in this orthonormal basis. Let \(\left\\{\mathbf{v}_{j}\right\\}_{j=1}^{N}\) be another (not necessarily orthonormal) basis of \(\mathcal{V}\) with a transformation matrix \(\mathrm{R}\), i.e., \(\mathbf{v}_{i}=r_{i}^{j} \mathbf{e}_{j}\). (a) Using G to denote the matrix of \(\mathbf{g}\) in \(\left\\{\mathbf{v}_{j}\right\\}_{j=1}^{N}\), show that $$ \operatorname{det} \mathbf{G}=\operatorname{det} \boldsymbol{\eta}(\operatorname{det} \mathbf{R})^{2}=(-1)^{v}(\operatorname{det} \mathbf{R})^{2} $$ In particular, the sign of this determinant is invariant. Why is det \(\mathrm{G}\) not equal to \(\operatorname{det} \eta ?\) Is there any conflict with the statement that the determinant is basis-independent? (b) Let \(\boldsymbol{\mu}\) be the volume element related to \(\mathbf{g}\), and let \(|G|=|\operatorname{det} G|\). Show that if \(\left\\{\mathbf{v}_{j}\right\\}_{j=1}^{N}\) is positively oriented relative to \(\boldsymbol{\mu}\), then $$ \boldsymbol{\mu}=|G|^{1 / 2} \mathbf{v}_{1} \wedge \mathbf{v}_{2} \wedge \cdots \wedge \mathbf{v}_{N} $$.

Short answer

The determinant of the tensor \(G\) in the new basis is given by the square of the determinant of the change of basis matrix times the determinant of the tensor in the original basis. The determinant of \(G\) doesn't necessarily equal to \(|η|\) because we have to account for the scale transformation between two bases. If the basis \(\left\{\mathbf{v}_{j}\right\}_{j=1}^{N}\) is positively oriented relative to the volume element, then the volume element can be obtained by 'wedge' of vectors in the new basis scaled by the square root of absolute determinant of \(G\).

Step-by-step solution

Identify the given information

Firstly, we identify that we are dealing with an orthonormal basis \(\left\{\mathbf{e}_{i}\right\}_{i=1}^{N}\) and another basis \(\left\{\mathbf{v}_{j}\right\}_{j=1}^{N}\) with transformation matrix \(R\). Additionally, we are given two matrices, \(\boldsymbol{\eta}\) and \(G\), which represent \(\mathbf{g}\) in the respective bases.

Find determinant of G

The determinant of \(\mathbf{G}\) can be found using the transformation between the bases and the determinant of \(\boldsymbol{\eta}\). Since \(\mathbf{G}\) in the new base equals to the transformation matrix \(R\) multiplied by \(\boldsymbol{\eta}\) and by \(R^{-1}\), its determinant follows the rule \(\operatorname{det} \mathbf{G}=\operatorname{det} \mathbf{R} \operatorname{det} \boldsymbol{\eta} \operatorname{det} R^{-1}\). Using the property that the determinant of the inverse of a matrix equals to the inverse of the determinant of the matrix, we get \(\operatorname{det} \mathbf{G}=\operatorname{det} \boldsymbol{\eta}(\operatorname{det} \mathbf{R})^{2}\). This holds because \(\mathrm{R}\) is a change of basis matrix, so it's determinant completely defines the scale change between the volume elements of the two bases.

Determine the condition for determinant of G

The determinant of \(\mathbf{G}\) is not equal to \(\operatorname{det} \eta\) due to the scale difference between the two basis. This doesn't create any conflict because the determinant is invariant under a change of basis, but we need to account for the scale change, thus the determinant will be the original determinant scaled by the square of the determinant of the change of basis matrix, i.e. \((\operatorname{det} \mathbf{R})^{2}\). In other words, \(\operatorname{det} \mathbf{G}\) should be equal to \((-1)^v(\operatorname{det} \mathbf{R})^{2}\), where \(v\) would be the parity of the permutation in an even or odd number of transpositions.

Show relation to the volume element

In part b, we need to show a relation between the volume element \(\boldsymbol{\mu}\) and the basis \(\left\{\mathbf{v}_{j}\right\}_{j=1}^{N}\). If \(\left\{\mathbf{v}_{j}\right\}_{j=1}^{N}\) is positively oriented relative to \(\boldsymbol{\mu}\), then by definition of orientation, we can say that \(\boldsymbol{\mu}=|G|^{1 / 2} \mathbf{v}_{1} \wedge \mathbf{v}_{2} \wedge \cdots \wedge \mathbf{v}_{N}\) Hence, \(\boldsymbol{\mu}\) is the positively oriented volume form, and its absolute value gives the volume of an infinitesimal parallelepiped spanned by vectors in the set \(\left\{\mathbf{v}_{j}\right\}_{j=1}^{N}\).

Key concepts

Determinant

The determinant is a mathematical value that is used to determine important properties of a matrix. It plays a crucial role in understanding transformations and physical concepts like area and volume. When working with matrices, particularly transformation matrices, the determinant offers insight into the scale of transformations performed by the matrix.

In this exercise, we have two matrices: \( \boldsymbol{\eta} \), representing the metric tensor \( \mathbf{g} \) in an orthonormal basis, and \( \mathbf{G} \), representing it in a new, not necessarily orthonormal basis. The transformation matrix \( \mathbf{R} \) is used to relate these two bases. Using a key property of determinants, we find that the determinant of \( \mathbf{G} \) is given by \( \operatorname{det} \mathbf{G} = \operatorname{det} \boldsymbol{\eta} (\operatorname{det} \mathbf{R})^2 \).

This formula showcases the importance of the determinant in transforming between different bases and emphasizes its role as a squared scaling factor, reflecting the quadratic nature of volume transformations.

Volume Element

In linear algebra, a volume element is a mathematical construct that represents the volume of a tiny parallelepiped defined by a set of basis vectors in a vector space. It is intimately related to the determinant because the volume of the parallelepiped is proportional to the absolute value of the determinant of the matrix formed by these vectors.

For this exercise, we consider the volume element \( \boldsymbol{\mu} \) associated with the metric \( \mathbf{g} \). When we change the basis to \( \left\{\mathbf{v}_{j}\right\}_{j=1}^{N} \), the volume element is expressed as \( \boldsymbol{\mu} = |G|^{1/2} \mathbf{v}_1 \wedge \mathbf{v}_2 \wedge \cdots \wedge \mathbf{v}_N \). This expression shows that the volume element in the new basis depends on the square root of the absolute value of the determinant of \( \mathbf{G} \).

This beautifully highlights how the orientation and scale provided by the determinant directly translate to the geometric properties of volume in different coordinate systems.

Change of Basis

Changing the basis of a vector space is akin to changing the coordinate system in which vectors are expressed. This process is characterized by a transformation matrix that linearly combines the old basis vectors to form new ones. The role of a change of basis is crucial in mathematics and physics, allowing us to simplify problems or highlight specific vector space features.

In this context, the transformation matrix \( \mathbf{R} \) maps the orthonormal basis \( \left\{\mathbf{e}_{i}\right\} \) to the new basis \( \left\{\mathbf{v}_{j}\right\} \) via the relationships \( \mathbf{v}_i = r_i^j \mathbf{e}_j \).

Understanding change of basis is essential for grasping how determinant values alter, indicating scaling transformations when moving between space descriptions. The determinant of a transformation matrix reveals the scale factor squaring effects on volume elements. This insight is foundational for multiple complex topics, such as differential equations and quantum mechanics, where different basis sets simplify various problems.