Business and industry leaders are often faced with discussions about the future of artificial intelligence (AI), machine learning, and data science. Similarly, they often engage in complex problem solving, for example, computer vision in robotics or semantic similarity in natural language processing (NLP) with implications for translation and speech recognition.

No matter how deep or shallow our understanding may be about the complexity of problems in all these hyped subject areas, we must sooner or later admit that we need to gain a better understanding of what lies at the heart of most problem-solving strategies: expressing linear relationships among objects, solving linear equations, and dealing with multivariate numeric data. All of these problem-solving approaches, however, boil down to working with vectors and matrices. Even when we deal with words in NLP, most successful algorithms for semantic similarity transform words into numbers and numerical data, for example, the Word2Vec approach. Moreover, developers of artificial neural networks (ANNs)-based approaches--for example, continuous bag-of-words (CBOW) for predicting a word from its context or the skip-gram model for predicting the context of a given word--will quickly find themselves facing co-occurrence matrices, weight matrices inside hidden layers, and their impact on activation functions, as well as facing problems related to the sheer size of matrices to be manipulated computationally.

In this context, questions about the transformation and factorization of the given matrices will inevitably arise, in order to save space and optimize the algorithms working with such matrices. We may also stumble upon key concepts such as principal component analysis (PCA), a matrix factorization technique that enables focusing on the very important features in ANN-based machine learning approaches, for instance, in object recognition.

Readers mainly interested in these or similar computational problems may jump directly to chapter 5. Readers interested in a broader context of how to apply matrices in data analysis and statistics can go straight to Part 2 (chapters 8 and 9). More specifically, Part 2 discusses various types of matrices as they are encountered in statistics. Readers interested in statistical computing and numerical linear algebra, as well as in the software and some algorithms behind the numerical methods for matrix manipulation, can go straight to Part 3 (chapters 10 through 12). Last but not least, we are left with the rest of Part 1 (chapters 1 to 7). This part covers the very fundamentals of vectors and matrices from a mathematical point of view. It offers a self-contained description of matrix algebra for applications in statistics.

This second edition may have a well-defined audience of graduate and advanced undergraduate students in, for instance, computational statistics and data science fields. However, AI and data science practitioners may also benefit from such a textbook. Regardless of the audience, one can hardly read this book in one go. It is most suitable as a reference book for when you need to understand one aspect or another of vectors, vector spaces, and matrices.

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