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This textbook demonstrates the strong interconnections between linear algebra and group theory by presenting them simultaneously, a pedagogical strategy ideal for an interdisciplinary audience. Being approached together at the same time, these two topics complete one another, allowing students to attain a deeper understanding of both subjects. The opening chapters introduce linear algebra with applications to mechanics and statistics, followed by group theory with applications to projective geometry. Then, high-order finite elements are presented to design a regular mesh and assemble the stiffness and mass matrices in advanced applications in quantum chemistry and general relativity.
This text is ideal for undergraduates majoring in engineering, physics, chemistry, computer science, or applied mathematics. It is mostly self-contained—readers should only be familiar with elementary calculus. There are numerous exercises, with hints or full solutions provided. A series of roadmaps are also provided to help instructors choose the optimal teaching approach for their discipline.

### Chapter 1. Vectors and Matrices

Abstract
What is a vector? It is a finite list of (real) numbers: scalars or components. In a geometrical context, the components have yet another name: coordinates. In the two-dimensional Cartesian plane, for example, a vector contains two coordinates: the x- and y-coordinates. This is why the vector is often denoted by the pair (xy). Geometrically, this can also be viewed as an arrow, leading from the origin (0, 0) to the point $$(x, y)\in \mathbb {R}^2$$. Here, $$\mathbb {R}$$ is the real axis, $$\mathbb {R}^2$$ is the Cartesian plane, and “$$\in$$” means “belongs to.”
Yair Shapira

### Chapter 2. Vector Product with Applications in Geometrical Mechanics

Abstract
How to use vectors and matrices? Well, we’ve already seen a few important applications: the sine, cosine, and Fourier transforms. Here, on the other hand, we use matrices and their determinant to introduce yet another practical operation: vector product in 3-D.
Yair Shapira

### Chapter 3. Markov Chain in a Graph

Abstract
So far, we’ve mostly used small matrices, with a clear geometrical meaning: $$2\times 2$$ matrices transform the Cartesian plane, and $$3\times 3$$ matrices transform the entire Cartesian space. What about yet bigger matrices? Fortunately, they may still have a geometrical meaning of their own. Indeed, in graph theory, they may help design a weighted graph, and model a stochastic flow in it. This makes a Markov chain, converging to a unique steady state. This has a practical application in modern search engines on the Internet [44].
Yair Shapira

### Chapter 4. Special Relativity: Algebraic Point of View

Abstract
To model static shapes in the plane, the ancient Greeks introduced Euclidean geometry. To model motion, on the other hand, Newton added a new time axis, perpendicular to the plane. This may help model a force, applied to an object from the outside to accelerate its original motion.
Yair Shapira

### Chapter 5. Group Representation and Isomorphism Theorems

Abstract
What is the most elementary algebraic object? This could be the individual number. In the previous part, we also introduced more complicated algebraic structures: vectors and matrices.
Yair Shapira

### Chapter 6. Projective Geometry with Applications in Computer Graphics

Abstract
What is a geometrical object? It is something that we humans could imagine and visualize. Still, in Euclidean geometry, a geometrical object is never defined explicitly, but only implicitly, in terms of relations, axioms, and logic ([22] and Chap. 6 in [63]).
Yair Shapira

### Chapter 7. Quantum Mechanics: Algebraic Point of View

Abstract
The matrices introduced above have two algebraic operations. Thanks to addition, they make a new linear space. Thanks to multiplication, they also form a group.
Yair Shapira

### Chapter 8. Polynomials and Their Gradient

Abstract
The polynomial is a special kind of function, easy to deal with. We start with a polynomial of just one independent variable: x. We discuss a few algebraic operations that can be applied to it. In particular, we introduce a few algorithms to calculate the value of the polynomial at a given argument.
Yair Shapira

### Chapter 9. Basis Functions: Barycentric Coordinates in 3-D

Abstract
Thanks to the above background, we are now ready to design a special kind of function: basis function (or B-spline). This will be the key to the finite-element method, with advanced applications in modern physics and chemistry.
Yair Shapira

### Chapter 10. Automatic Mesh Generation

Abstract
Consider a complicated domain in three spatial dimensions. How to store it on the computer? For this purpose, it must be discretized: approximated by a discrete mesh, ready to be used in practical algorithms.
Yair Shapira

### Chapter 11. Mesh Regularity

Abstract
In Chap. 10, Sects. 10.​1.​310.​1.​4, we’ve already met the important concept of mesh regularity, and took it into account in the refinement step. Here, we continue to discuss it, and introduce a few reliable tests to estimate it. This way, we can make sure that our multilevel refinement is indeed robust: the meshes are not only more and more accurate but also fairly regular.
Yair Shapira

### Chapter 12. Numerical Integration

Abstract
Does our multilevel refinement work well? Does it approximate well the original domain? To check on this, we use numerical integration.
Yair Shapira

### Chapter 13. Spline: Variational Model in Three Spatial Dimensions

Abstract
Once our mesh is sufficiently regular and accurate, basis functions (B-splines) can be defined in it. What is a basis function? It has the following properties.
Yair Shapira

### Chapter 14. Quantum Chemistry: Electronic Structure

Abstract
Let’s see how linear algebra and group theory can combine in a practical application in quantum chemistry: the electronic structure in an atom or a molecule. Indeed, the position of each electron is a random variable: we can never tell it for sure, but only at some probability. Likewise, energy and momentum are nondeterministic as well: we can never know what they are precisely, but only with some uncertainty.
Yair Shapira

### Chapter 15. General Relativity: Einstein Equations

Abstract
Here is an interesting application in general relativity: Einstein equations.
Yair Shapira