In this chapter, we present a few selected subjects that are important in applications as well but are not usually included in a standard linear algebra course. These subjects may serve as supplemental or extracurricular materials. The first subject is the Schur decomposition theorem, the second is about the classification of skew-symmetric bilinear forms, the third is the Perron–Frobenius theorem for positive matrices, and the fourth concerns the Markov or stochastic matrices.

The loss of human life and physical injuries through violence are an inherent consequence of armed conflict, including civil wars. Deliberate atrocities – such as war crimes, crimes against humanity, genocide, politicide and “ethnic cleansing” – have been a conspicuous feature of many wars. Civil wars – whether correctly or incorrectly from an empirical perspective – have often been regarded as particularly vicious, transgressing all norms of decency in the frequency and type of atrocities. This chapter explores several key questions that have arisen in the conflict analysis field in relation to atrocities in civil war – and war generally. Are atrocities specifically associated with certain “types” of civil war, such as separatist, ideological, intercommunal, or resource conflict? Are there patterns in terms of which types of actors – state or non-state rebel groups – are more likely to perpetrate atrocities? What motivates individuals and groups to perpetrate atrocities, and what “role,” if any, do such atrocities play in armed conflict? Do atrocities play a strategic role, or are they better understood as a manifestation of individual and group sadism, revenge, and hate or fear, spread in the contemporary era by social media? Are all combatants capable of perpetuating atrocities in the “right” circumstances? The chapter concludes with a discussion of the international norms that have emerged over the last century – which prohibit war crimes, crimes against humanity, and genocide – and the calls for accountability and justice after mass atrocities that have arguably made a significant although limited impact on conduct in war. As a part of this, “transitional justice” has emerged as an important topic, designed to address the societal impact and legacy of atrocities.

A rich and important area for the applications of linear algebra is machine learning. In machine learning, one aims to achieve optimized or learned understanding of various kinds of real-world phenomena from data collected or observed, without real comprehension of the functioning mechanisms of such phenomena. These functioning mechanisms are often impossible or unpractical to grasp anyway. In this chapter, we present several introductory and fundamental problems in supervised machine learning including linear regression, data classification, and logistic regression and the mathematical and computational methods associated.

In East Asia, the liberal Westernizing tendencies of the 1920s were replaced in the 1930s by authoritarian single-party rule in China and ultranationalistic militarism in Japan. Japan was wracked by a series of assassinations and attempted coups, which left the miliary in control. On the pretext of a staged explosion on the tracks of the Japanese-run South Manchurian Railroad (in China) in 1931, the Japanese army seized control over much of Manchuria and established a puppet state called Manchukuo. While Chiang Kai-shek struggled to put the Republic of China on a secure foundation, the rising communist leader Mao Zedong began experimenting with rural peasant revolution. After Chiang Kai-shek was kidnapped by his own generals and compelled to agree to a United Front with the Communists against Japan, a minor incident in July 1937 triggered the start of full-scale war with Japan. Japan’s inability to decisively defeat Nationalist China, then, led Japan to expand the war, eventually attacking Pearl Harbor and bringing the Allies into the war on China’s side.

Chapter 2 explores how racial classifications within statutory law affected who gets to vote, immigrate to the United States, self-govern, or become citizens in the years after the Fourteenth Amendment. The chapter addresses current questions related to immigration and paths to citizenship, and also focuses on the criteria that have been used to determine both the qualifications for citizenship and the rights and obligations that come with it. Sovereignty and the ability to self-govern is the core question discussed in relation to indigenous peoples. The ways in which whiteness shaped the definition and exercise of citizenship and sovereignty are introduced, emerging more fully in subsequent chapters. The debates explored here demonstrate the centrality of the consistent demand for assimilation into “American culture,” with the concomitant justification of excluding those who cannot match our expectations of citizenship due to stereotypes of their culture and “race.” Students will reckon with the significance and intention behind the framing of a singular “American culture,” and the ongoing demands for and costs of racial and cultural assimilation.

The East Asian community of states, having matured by the tenth century, thereafter continued to develop along somewhat independent trajectories. China settled into a distinctive late imperial form, characterized by the presence of an examination-recruited mandarin elite. Korea became more uncompromisingly Confucian; Vietnam gained independence; and Japan came to be dominated by uniquely Japanese warrior elites: the Shogun and samurai. China, under the Song Dynasty, became a major center of maritime trade and advanced technology. In the thirteenth century the Mongols erupted into East Asia, conquering China and reducing Korea to a subordinate “son-in-law” state. The Mongols, ironically, helped expose Koreans to Neo-Confucianism, and under increasing Confucian influence, Chinese-style family patterns became the new “tradition” in Korea. In Japan, conflict between leading warrior families resulted in the epochal Gempei War (1180–1185), from which Minamoto Yoritomo emerged victorious and became Japan’s first Shogun. By the end of this period, growing volumes of maritime trade ushered East Asia into the Early Modern age.

The Introduction of the book exposes students to key concepts surrounding how U.S. law has constructed and maintained the idea of race in society. Introducing the framework of individual and systemic oppression, the chapter explores the dualistic role that the law plays in the democratic process by both advancing and preventing societal change, based on how it is wielded.