Practical linear algebra
"Practical Linear Algebra covers all the concepts in a traditional undergraduate-level linear algebra course, but with a focus on practical applications. The book develops these fundamental concepts in 2D and 3D with a strong emphasis on geometric understanding before presenting the general (n-dimensional) concept. The book does not employ a theorem/proof structure, and it spends very little time on tedious, by-hand calculations (e.g., reduction to row-echelon form), which in most job applications are performed by products such as Mathematica. Instead the book presents concepts through examples and applications. "
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5recommend
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What Hacker News says
- For my fellow visual thinkers who might be looking for a linear algebra book that focuses more on developing the geometric intuition for stuff like this, rather than just pure numeric linear system solving, let me recommend:
"Practical Linear Algebra: A Geometry Toolbox" by Farin and Hansford.
- You might want to checkout the book Practical Linear Algebra: A Geometry Toolbox by Dianne Hansford and Gerald Farin (its 1st edition was simply named The Geometry Toolbox: For Graphics and Modeling) to get an intuitive and visual introduction to Linear Algebra.
Pair it with Edgar Goodaire's Linear Algebra: Pure & Applied and you can transition nicely from intuitive geometric to pure mathematical approach. The author's writing style is quite accessible.
Add in Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares by Stephen Boyd et al. and you are golden. Free book available at https://web.stanford.edu/~boyd/vmls/
- What you are trying to learn/understand falls under the rubric of Scientific Computing/Numerical Methods/Numerical Analysis/Numerical Algorithms. The mathematics underpinning them is quite wide but mostly Linear Algebra and Calculus. You might find the following useful for your study;
1) Scientific Computing by Michael Heath - Classic text covering a broad swath of domains and tries to build motivation/intuition before the mathematics (affordable Indian edition available).
2) Mathematical Principles for Scientific Computing and Visualization by Gerald Farin and Dianne Hansford - Nice overview of needed background. The authors also have a book named Practical Linear Algebra: A Geometry Toolbox which you might find a ideal companion.
3) Numerical Methods: Fundamentals and Applications by Rajesh Kumar Gupta - A relatively recent book with a really broad coverage of subjects and detailed mathematical expositions (affordable Indian edition available).
4) Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics by Justin Solomon - Good explanations (affordable Indian edition available). Free ebook available at https://people.csail.mit.edu/jsolomon/
5) Finally, Mathematics for Physicists: Introductory Concepts and Methods by Alexander Altland and Jan Von Delft is an excellent book to have as a reference. It has three sections viz. Linear Algebra, Calculus and Vector Calculus. The presentation is very precise, does not focus on proofs/lemmas but on concepts and covers a wide swath of important mathematics.
- You are just placing too much emphasis on caveats and limitations rather than focusing on the usefulness of the technique itself. Also techniques like Dimensionality Reduction exist to help tame higher dimensions.
None of your cautions are really that big of an impediment in practice and i would not want students to take away the wrong message from these comments. Mathematicians/Practitioners know when to switch from simple geometry to algebra and that is taught as a matter of course.
As an example, the farin and hansford book Practical Linear Algebra: A Geometry Toolbox that i mention at https://news.ycombinator.com/item?id=45060305 precisely takes this approach starting with visualizing vectors in 2D/3D geometry and then moving on to higher-dimensions. Its 1st edition was actually named The Geometry Toolbox: For Graphics and Modeling.
- You might want to watch this short video a couple of times - https://www.youtube.com/watch?v=rowWM-MijXU
A simple way to think about Linear Algebra is as a set of rules i.e. functions (aka Matrices) applied to input points (aka vectors) to give output points in a vector (i.e. multidimensional) space. Thus multi-valued inputs give multi-valued outputs. The key is to keep both the algebraic manipulations and what they may mean in the geometric interpretation simultaneously in mind. The basic example of solving a system of linear equations in two variables will cement this idea when you map it to the XY coordinate system.
You might find the book Practical Linear Algebra : A Geometry Toolbox by Farin and Hansford useful for further study.
- You need this book: Practical Linear Algebra: A Geometry Toolbox by Gerald Farin and Dianne Hansford. You can get older editions for pennies. The 1st edition was actually called The Geometry Toolbox for Graphics and Modeling.
You should also read Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares by Stephen Boyd and Lieven Vandenberghe: https://web.stanford.edu/~boyd/vmls/
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