Application of Linear Algebra to Physics

[azupol]

I've just begun my 2nd year of study at my University and as part of a program requirement, must take a Liner Algebra course. I love mathematics, but I have never experienced much of Linear Algebra. I was just wondering how Linear Algebra applies to Physics, what branches, etc.

 

[TheFerruccio]

The first thing that came to mind for me was the use of Hilbert spaces in quantum mechanics.

Linear algebra also has applications in designing feedback control systems, dealing with something called "state space."

I extensively used linear algebra when dealing in aerospace structures classes, representing stresses and strains as a matrix.

This PDF might explain a bit in more detail:

http://www.math.uni-bielefeld.de/~hemion/Linear_Algebra_in_Physics/LinearAlg.Physics.pdf

It's pervasive and entirely necessary. You'll be using it all the time, and it makes a bunch of mathematics a whole lot easier.

 

[HallsofIvy]

I would take the "Hilbert spaces in Quantum mechanics" with a grain of salt. "Linear Algebra", strictly speaking, deals with finite dimensional vector spaces while Hilbert spaces are infinite dimensional- the domain of "functional analysis". Of course, one had better have a good grounding in Linear Algebra before starting "functional analysis".

 

[azupol]

Thanks for your responses and the PDF, it was very helpful and descriptive. I suppose my next question would be, what would be some math courses you guys would recommend I take to further my understanding of Physics? Along with Linear Algebra, I am also taking Multivariable Calculus, Differential Equations, Concepts in Abstract Mathematics, and Complex Analysis.

 

[blue_raver22]

Linear algebra is a fundamental tool in the field of physics. It is used to describe and analyze physical systems, from classical mechanics to quantum mechanics. In classical mechanics, linear algebra is used to represent the motion of objects and the forces acting on them. It is also used to solve systems of equations and find solutions to differential equations, which are essential in understanding the behavior of physical systems.

In quantum mechanics, linear algebra plays a crucial role in describing the behavior of particles and their interactions. The fundamental principles of quantum mechanics, such as superposition and entanglement, are described using linear algebra concepts. Additionally, matrix operations and eigenvectors are used to calculate the energy levels and probabilities of quantum systems.

Other branches of physics, such as electromagnetism and thermodynamics, also heavily rely on linear algebra. In electromagnetism, linear algebra is used to describe the behavior of electric and magnetic fields, while in thermodynamics, it is used to analyze the flow of heat and energy in a system.

Overall, linear algebra is a powerful tool that is essential for understanding and solving complex problems in physics. As you continue your studies, you will see how it is applied in various branches of physics and how it helps to uncover the underlying principles that govern the physical world. I hope you enjoy your linear algebra course and see its relevance in physics.