What Are the Real-World Applications of Linear Transformations?

[EvLer]

applications ?

We are studying linear transformations right now in my Lin. Alg. class. And I like to think that mathematics has some application in the real world. But what kind of appliation do matrix transfomations have? Are there any algorithms based on it? If not, it's kind of pointless in and of itself

 

[matt grime]

Linear systems are pretty much the only ones that we can always solve, if you really need to think in terms of actual physical things that need linear algebra.

When you have more than one variable to keep track of then you need matrices, or linear maps.

Dynamical systems, quantum mechanics, epidemiology, machine learning, computing, weather forecasting, absolutely anything that has equations in it will require, at some level, a knowledge of linear algebra.

 

[PerennialII]

Yeah, they're all over the place. A huge fraction of worlds computing capacity is spent working those matrices, in the end most physical problems reduce to a "simple" matrix equation in need of solving.

 

[matt grime]

And, although thinking about them as "transformations" of space might seem unrelated to their use in these other situations, the geometrical intuition you develop can help you, particularly when looking for eigenvalues, eigenvectors and eigenspaces and interpreting their meaning. For instance in a simple 2-dimensional system

x_n = Ax_{n-1}

an eigenvector of eigenvalue 1 corresponds to a fixed point, and other orbits and limits can be interpreted as stable or unstable and so on.