Question

Show that \(G L(\mathcal{V})\) is not a compact group. Hint: Find a continuous function \(G L(\mathcal{V}) \rightarrow \mathbb{C}\) whose image is not compact.

Short answer

The general linear group \(G L(\mathcal{V})\) is not compact, which has been proved through demonstrating the existence of a continuous function from \(G L(\mathcal{V})\) to complex numbers whose image is not compact.

Step-by-step solution

Define the continuous function

Start by defining a continuous function from \(G L(\mathcal{V})\) to the complex numbers, \(\mathbb{C}\). Let \(f: G L(\mathcal{V}) \rightarrow \mathbb{C}\) be defined as \(f(A) = \text{tr}(A)\), where \(A\) is any element in \(G L(\mathcal{V})\), 'tr' denotes the trace of the matrix \(A\), and the image of \(f\) is the set of all complex numbers 'c' where c is the trace of some \(A \in G L(\mathcal{V})\). This function, \(f\), is continuous.

Show that \(f(A)\) is not compact

Now, we need to show that the image set of the function \(f\), is not compact. The trace of a matrix \(A\) is the sum of the elements on the main diagonal. Because \(A \in G L(\mathcal{V})\), the size of \(A\) could be infinitely large, therefore the sum of the elements on the main diagonal could be infinitely large. This indicates that the set of all possible values of \(f(A)\) is not bounded, it covers the entire field of complex numbers. Therefore, the image of function \(f\), i.e., the set \(f(A)\) is not compact as it's not bounded.

Use the principle of continuous image of a compact set

Recall the principle that 'continuous image of a compact set is compact'. Here, since we've shown that \(f(A)\) is not compact, it means that \(G L(\mathcal{V})\) cannot be compact.

Conclusion

Therefore, using the facts above and theorem of continuous image of compact set, we conclude that, \(G L(\mathcal{V})\), the general linear group, is not a compact group.

Key concepts

General Linear Group

In the context of linear algebra and geometry, the general linear group, denoted as \(GL(V)\), is a fundamental mathematical structure. It comprises all invertible matrices on a vector space \(V\) with entries that can be either real or complex numbers. The 'invertibility' means that for every matrix \(A\) in \(GL(V)\), there exists another matrix \(B\) such that \(AB = BA = I\), where \(I\) is the identity matrix.

Being able to operate with such matrices is critical for solving systems of linear equations, finding determinants, and understanding geometric transformations. When we discuss whether this group is 'compact', we are essentially exploring whether it can be contained within a finite and closed region in complex or real space. For groups like \(GL(V)\), this property is related to the bounds (or lack thereof) on the size of the matrices' entries, which is essential for understanding the behavior of transformations in infinite-dimensional spaces.

Continuous Function

A continuous function is one where small changes in the input result in small changes in the output; there are no abrupt jumps or breaks. Within the realm of complex functions, which map elements between spaces, continuity ensures a predictable and smooth transition between points.

Mathematically, for a function \(f: X → Y\), if for every point \(x\) in the space \(X\) and every positive number \(ε\), there is a positive number \(δ\) such that every point \(x'\) within a distance \(δ\) of \(x\) has an \(f(x')\) within a distance \(ε\) of \(f(x)\), then \(f\) is continuous. This concept is key to many parts of mathematical analysis and topology because the behavior of continuous functions over compact sets has elegant properties, such as the ability to be uniformly continuous, bounded, and to attain a maximum or minimum.

Complex Numbers

Complex numbers are an extension of the real numbers and are formidable tools in both engineering and scientific computation. A complex number is usually expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\). The set of all complex numbers is denoted by \(\mathbb{C}\).

They afford a great deal of flexibility in mathematics, allowing the solution of equations that no real number can solve, such as \(x^2 + 1 = 0\). Complex numbers can be plotted on a plane, with the 'real' part \(a\) on one axis and the 'imaginary' part \(b\) on the other. This 'complex plane' grants us a more nuanced view of mathematical behavior and provides a natural environment for talking about the convergence of sequences and the compactness of sets in higher-dimensional spaces.

Trace of a Matrix

The trace of a matrix is a concept in linear algebra that can be surprisingly powerful. It is defined as the sum of the elements on the main diagonal of a square matrix. For a matrix \(A\), with elements \(a_{ii}\) on the main diagonal, the trace is denoted as \(\text{tr}(A)\) and is calculated by \(\text{tr}(A) = \sum_{i} a_{ii}\).

The trace has several important properties. It is invariant under change of basis, meaning it does not depend on the coordinate system used to describe the matrix. Also, the trace is linear, meaning that the trace of the sum of two matrices equals the sum of their traces. Moreover, the trace helps in the study of eigenvalues as it equals the sum of the eigenvalues of \(A\). These attributes make the concept of the trace a pivotal tool for mathematicians and scientists in many fields, including quantum mechanics and matrix theory.