In this chapter, we exclusively consider vector spaces over the field of reals unless otherwise stated. First, we present a general discussion on bilinear and quadratic forms and their matrix representations. We also show how a symmetric bilinear form may be uniquely represented by a self-adjoint mapping. Then we establish the main spectrum theorem for self-adjoint mappings based on a proof of the existence of an eigenvalue using Calculus. Next we focus on characterizing the positive definiteness of self-adjoint mappings. After these we study the commutativity of self-adjoint mappings. As applications, we show the effectiveness of using self-adjoint mappings in computing the norm of a mapping between different spaces and in the formalism of least squares approximations.

Language use involves the activation of phonological, morphological, grammatical and lexical systems for meaning-making with other people in specific contexts. Therefore, we not only need to acquire and develop these linguistic systems for language use, but we also need to develop an awareness and understanding of these linguistic systems as meaning-making resources for appropriate use in a given context. For this reason, it is necessary to focus on the social use of language as a key aspect of language development.

In this chapter, we extend our study of linear algebraic structures to multilinear ones that have broad and profound applications beyond those covered by linear structures. First, we give some remarks on the rich applications of multilinear algebra and consider multilinear forms in a general setting as a starting point that directly generates bilinear forms already studied. Next, we specialize our discussion to consider tensors and their classifications. Then, we elaborate on symmetric and antisymmetric tensors and investigate their properties and characterizations. Finally, we discuss exterior algebras and the Hodge dual correspondence.

Readers will understand the physical laws that form the basis of the fluid equations of motion, and will learn how to obtain the equations of fluid motion in both derivative and integral form. Presentations are included to show how to apply the equations of motion to calculate properties of fluid flows. Readers will understand dynamic similarity and how to calculate Mach number and Reynolds number, including descriptions of the various Mach and Reynolds number regimes and their distinguishing characteristics.

In this chapter, we extend our study on real quadratic forms and self-adjoint mappings to the complex situation. We begin by a discussion on the complex version of bilinear forms and the Hermitian structures. We will relate the Hermitian structure of a bilinear form with representing it by a unique self-adjoint mapping. Then we establish the main spectrum theorem for self-adjoint mappings. Next we focus again on the positive definiteness of self-adjoint mappings. We explore the commutativity of self-adjoint mappings and apply it to obtain the main spectrum theorem for normal mappings. We also show how to use self-adjoint mappings to study a mapping between two spaces.

This chapter highlights the knowledge required to work with diverse students who communicate using the different varieties of English that exist in Australia. In line with the ‘Language variation and change’ sub-strand of the Australian Curriculum: English, we discuss linguistic and cultural diversity through the concept of plurilingualism, and the transcultural and sociolinguistic competence and knowledge required by teachers working with culturally and linguistically diverse (CALD) learners. We highlight the challenges and rewards associated with instructing students from varying linguistic and cultural backgrounds. We also stress the crucial role teachers play in nurturing learners of English as an additional language or dialect (EAL/D) students.

Understanding the causes of intrastate armed conflict and civil wars – whether as individual cases or in a more general sense – is the most compelling but perhaps also the most elusive challenge in the study of such conflicts. In this field, causal relationships are complex and difficult to establish beyond doubt, and discrete direct causes rarely exist. This chapter explores the methodological challenges that arise when seeking to identify direct or indirect causes of civil wars, in particular across multiple cases. It presents key theories of civil war onset in relation to political, economic, social, institutional, ecological, identity, and governance conditions. It gives particular attention to “greed” and “grievance” as key concepts for understanding why intrastate armed conflicts occur, the association between democratization and increased risk of violent conflict, and the concept of “ethnic conflict” as a cause of civil war. The question of whether it is more helpful to focus on enabling factors – the conditions that allow violent uprisings to occur – or motivations for participating in armed conflict to understand the causes of civil war is also discussed. The chapter concludes by considering the implications these debates raise for policies designed to prevent violent conflict and build sustainable peace.

Spoken language consists of a complex system of sounds that infants first learn to perceive and produce through social interaction in their linguistic community. This initial exposure before school is critical in preparing them for school, where they learn to encode their spoken language into written language. Once children can sufficiently speak and understand most language at around the age of five, it becomes the role of educators, and parents/care providers to assist them in the development of their explicit phonological awareness. Phonological awareness, which refers to the broad understanding of the sound structure of language, plays a critical role in the development of reading and writing. For instance, before children can effectively start learning to read, they need to develop an awareness of the sound structure of words, including syllables, rhymes and individual speech sounds called phonemes, as well as the ability to segment and blend phonemes.

In this chapter, we present two important and related problems in data analysis: the low-rank approximation and principal component analysis (PCA), both based on singular value decomposition. First, we consider the low-rank approximation problem for mappings between two vector spaces. Next we specialize on the low-rank approximation problem for matrices in both induced norm and the Frobenius norm, which are of independent interest for applications. Then we consider PCA. These results are also useful in machine learning. Furthermore, as an extension of the ideas and methods, we present a study of some related matrix nearness problems.

Consolidate what you learned in Chapters 7–10; revisit the ideas you came across in Chapters 7–10 in a cross-cultural context.