Welcome to Linear Algebra! This branch of mathematics deals with vector spaces, linear transformations, and systems of linear equations. It has numerous applications in computer graphics, machine learning, engineering, physics, and economics. In this visualization, we can see vectors represented as arrows in a coordinate system. The blue and red vectors can be added together to form the green vector, demonstrating one of the fundamental operations in linear algebra: vector addition.
Matrices are rectangular arrays of numbers arranged in rows and columns. They are powerful mathematical objects that can represent linear transformations. A linear transformation maps vectors from one vector space to another while preserving vector addition and scalar multiplication. Common transformations include rotations, scaling, reflections, and shearing. Here, we see a square being rotated by 45 degrees. This rotation can be represented by a 2x2 matrix shown below the grid. When we multiply this matrix by the coordinates of each point in the original square, we get the coordinates of the corresponding points in the rotated square.
Systems of linear equations are a fundamental topic in linear algebra. A system of linear equations can be represented using matrices, which provides a compact way to work with multiple equations simultaneously. There are several methods to solve these systems, including Gaussian elimination, matrix inversion, and Cramer's rule. In this example, we have a system of two equations with two unknowns. We can solve it algebraically by converting to an augmented matrix and performing row operations. Geometrically, each equation represents a line in the coordinate plane, and the solution is the point where these lines intersect. Here, the solution is x equals 18/7 and y equals 22/7.
Eigenvalues and eigenvectors are among the most important concepts in linear algebra. For a square matrix A, an eigenvector is a non-zero vector v such that when A is multiplied by v, the result is a scalar multiple of v. This scalar is called the eigenvalue. Mathematically, we write this as A times v equals lambda times v, where lambda is the eigenvalue. In this visualization, we have a 2 by 2 matrix with two eigenvectors shown in blue and red. When the matrix transforms these vectors, they only change in length, not direction. Eigenvalues and eigenvectors have numerous applications, including principal component analysis in statistics, quantum mechanics in physics, vibration analysis in engineering, and even Google's PageRank algorithm for ranking web pages.
Let's summarize what we've learned about linear algebra. First, vectors are mathematical objects with both magnitude and direction, forming the foundation of linear algebra. Second, matrices represent linear transformations and systems of equations, allowing us to perform complex operations efficiently. Third, systems of linear equations can have zero, one, or infinitely many solutions, depending on the relationship between the equations. Fourth, eigenvalues and eigenvectors reveal fundamental properties of linear transformations, making them crucial in many applications. Finally, linear algebra has applications in virtually all fields of science and engineering, from computer graphics and machine learning to quantum physics and economics. Understanding these concepts provides powerful tools for solving a wide range of problems.