Education

Abstract

This issue of SIAM Review contains two papers in the Education section. The first, “A Generalized Dual Transform: Linear Algebra and Geometry of (Pseudo)Inverting a Matrix,” is presented by L. P. Withers, Jr. For a linear subspace $A$ of a vector space $V$, we may have a nonorthogonal basis of $A$. We could obtain an orthogonal basis (e.g., by the Gram--Schmidt orthogonalization procedure) and the orthogonality helps us to represent each element $b\in A$ as a linear combination of the new basis in an easy way. How should we express $b$ as a linear combination of the original, unorthogonalized vectors? One suggestion is to construct a complementary list of vectors, called a dual list, such that each pair of vectors $a^i,a^j$ on that list are orthogonal and each vector has a length of one. The construction is called the dual transform. Involving the complementary subspace of $A$ in $V$ and orthogonal projections, we obtain again a simple formula for the representation of $b$. Next to generating dual vectors, the Gram--Schmidt orthogonalization procedure exhibits other interesting properties, which lead to a parallel so-called butterfly process for computing the dual transform. The article proceeds to explain how the dual transform is generalized via axioms and how the respective procedures are performed in the general setting. Several examples supplement the discussion. The paper is accessible to advanced undergraduate students with basic knowledge in linear algebra and complex analysis. The second article, “When Randomness Helps in Undersampling,” was written by Roel Snieder and Michael B. Wakin. In our digital era, we use many recordings: music, sounds of nature, and others. Many other signals such as telecommunication, temperature, and air pressure, are recorded for practical and scientific purposes. When signals are stored in computers, they are digitized by collecting and storing values of some functions at discrete times. A straightforward thought on how to accomplish that is to collect values uniformly in time and in all frequency components. However, the measurements might be feasible only for some times or frequencies; otherwise data acquisition or data transmission of a full collection might be too burdensome. When some times or frequencies are left out, it is said that the signal is undersampled. In that case, it is best to choose the times or frequencies to be left out randomly instead of uniformly. The authors focus on the problem of reconstructing a signal in the time domain using undersampled frequency components. The benefits of random undersampling are illustrated with an example of the air pressure recorded at a volcano in Costa Rica, but the authors cite other sources on seismic surveys as well as magnetic resonance imaging where benefits from undersampling are evidenced. The key to understand the phenomena is to analyze the effect of the sampling on the discrete Fourier transform, which allows a discrete-time signal of length $N$ to be represented as a sum of $N$ complex exponential terms. The authors have made the signal processing codes and the data that are used for the article available at https://mines.edu/~mwakin/software. The paper is directed toward undergraduate students in engineering with a mathematical background and interests.