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*Linear Algebra for the 21st Century*via either Oxford University Press, or via Amazon.This book establishes a new route through linear algebra, one that develops early the Singular Value Decomposition (SVD), the so-called "jewel in crown" of linear algebra. Thereafter the beautiful power of the SVD both explores many modern applications, especially in Data Science, and also develops traditional linear algebra concepts, theory, and methods. No rigour is lost in this new route: indeed, this book demonstrates that most theory is better proved with an SVD rather than with a traditional approach. This new route through linear algebra becomes available by the ready availability of ubiquitous computing in the 21st century.

## Three important features

- Integrates the availability of ubiquitous computing into linear algebra.
- This meets the demands made in recent national reviews of undergraduate mathematics curricula (e.g., Alpers et al. 2013, Bressoud et al. 2014, Turner et al. 2015, Schumacher et al. 2015, Bliss et al. 2016 )
- Reforms/reshapes linear algebra to best suit 21st century teaching, theory and application.
- This is especially suitable for the so-called `big data' of bioinformatics, data mining, image processing, and artificial intelligence.
- The Singular Value Decomposition (SVD) is a key enabling tool.
- The SVD is sometimes called the jewel in the crown of linear algebra. Limitations of traditional by-hand methods make the SVD inaccessible, but with ubiquitous computing this book utilises its power for both applications and theory.

## The future of linear algebra

The book is aimed for students in the first year or two of university study of the mathematics of linear algebra for science, engineering or computing. It also forms an ideal companion to supplement modern statistics courses. Much of the design of traditional mathematics curricula was predicated on `by hand' calculation. Ubiquitous computing via laptops, iPads and smart-phones requires us to refresh what we teach and how it is taught, especially as computing has become the `third arm' of engineering and science. This book reforms linear algebra to best suit 21st century teaching and application, especially for the so-called `big data' of bioinformatics, data mining, image processing, and artificial intelligence. Just as importantly, this reform improves the learning of the theory of linear algebra. With ubiquitous computing the approach to linear algebra needs to be reimagined. Recent national reviews of curricula identified the need (Alpers et al. 2013, Bressoud et al. 2014, Turner et al. 2015, Schumacher et al. 2015, Bliss et al. 2016, e.g.), and this book answers the call. Modern applications of linear algebra, especially in the rapidly broadening fields of Data Mining and Artificial Intelligence, and also in fields such as Bioinformatics, require the Singular Value Decomposition (SVD). This book places the SVD central and early to empower the students in these disciplines to learn and use the best techniques. Indeed the SVD is sometimes called the jewel in the crown of linear algebra, and this book makes its power available early and broadly to students and courses. No rigour is lost in this new route: indeed, this book demonstrates that most theory is better proved with an SVD. Additionally, there is earlier introduction, development and emphasis on orthogonality that is vital in so many applied disciplines throughout science, engineering, computing and increasingly social sciences. A strong graphical emphasis takes advantage of the power of visualisation in the human brain. Examples throughout include modern applications such as GPS, text mining, image processing. Active learning exercises throughout are aimed to enhance lectures, quizzes, or `flipped' teaching.As so many other disciplines use the SVD, it is not only important that mathematicians understand what it is, but also teach it thoroughly in linear algebra (Turner et al., 2015, p.30)

## LaTeX sources for graphic exercises

To help teachers generate good graphical questions, here are links to LaTeX source code for generating graphical exercises corresponding to ones in the book.**Vectors**- None.

**Systems of linear equations****Matrices encode system interactions**- 3.2.11 which appear to correspond to multiplication by a diagonal matrix ...
- 3.2.12 which appear to correspond to multiplication by a diagonal matrix ...
- 3.2.13 Draw approximate orthogonal coordinate axes ...
- 3.2.15 vectors appear to form an orthogonal set ...
- 3.2.16 vectors appear to form an orthogonal set ...
- 3.2.24 which appear to be that of multiplying by an orthogonal matrix ...
- 3.2.25 which appear to be that of multiplying by an orthogonal matrix ...
- 3.5.21 draw the orthogonal projection ...
- 3.5.28 draw its orthogonal complement ...
- 3.5.31 draw the decomposition of a vector into the sum ...
- 3.6.1 which cannot be that of a linear transformation ...
- 3.6.3 which cannot be that of a linear transformation ...
- 3.6.18 estimate the standard matrix of the linear transformation ...

**Eigenvalues and eigenvectors of symmetric matrices****Approximate matrices****Determinants distinguish matrices****Eigenvalues and eigenvectors in general**

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