Mathematics Describing the Real World: Precalculus and Trigonometry Season 1 Episode 28 Applications of Linear Systems and Matrices

In season 1 episode 28 of Mathematics Describing the Real World: Precalculus and Trigonometry, viewers will explore the practical applications of linear systems and matrices. The episode begins by introducing the concept of linear systems, which involve a set of equations that can be represented graphically as lines intersecting at a point. Linear systems are used in many real-world situations, such as calculating the optimal amount of ingredients in a recipe or determining the most efficient way to travel between multiple destinations.

The episode then delves into the properties of matrices, which are arrays of numbers that can be used to represent linear systems. Viewers will learn about the different operations that can be performed on matrices, including addition, subtraction, and multiplication. They will also discover how matrices can be used to solve systems of equations, and how these solutions can be checked using matrix multiplication.

One of the primary applications of linear systems and matrices is in computer graphics, and the episode explains how these concepts are used to create images and animations. Viewers will learn about the various transformations that can be applied to objects in a scene, such as translation, scaling, and rotation, and how these transformations can be represented using matrices. They will also see how linear systems can be used to simulate the physics of objects, such as calculating the trajectory of a projectile.

The episode also explores some of the ways that linear systems and matrices are used in economics. Viewers will learn about input-output models, which are used to calculate how changes in one sector of the economy will affect other sectors. They will see how these models can be represented using matrices, and how they can be used to make predictions about how different policies will impact the economy.

Finally, the episode looks at some of the more theoretical aspects of linear systems and matrices. Viewers will learn about eigenvectors and eigenvalues, which are special vectors and numbers that can be used to simplify computations involving matrices. They will also see how these concepts are used in quantum mechanics, where they are used to represent the wave functions of particles.

Overall, this episode of Mathematics Describing the Real World: Precalculus and Trigonometry offers a comprehensive overview of the practical uses of linear systems and matrices in a variety of fields. Viewers will come away with a deeper understanding of these concepts and how they can be applied in real-world situations.