Linear algebra is a fundamental branch of mathematics that deals with vectors, matrices, and linear transformations. It provides the mathematical foundation for many areas including computer graphics, machine learning, and engineering. In this introduction, we'll explore the basic concepts starting with vectors in a coordinate system.
Vector operations are fundamental in linear algebra. Vector addition can be visualized using the parallelogram rule. When we add two vectors, we place them head to tail, and the sum is the vector from the origin to the final point. This creates a parallelogram where the diagonal represents the sum of the two vectors.
Matrices are powerful tools that represent linear transformations. A two by two matrix transforms vectors by mapping the standard basis vectors i-hat and j-hat to new positions. The columns of the matrix tell us where these basis vectors land after the transformation. This completely determines how any vector in the plane will be transformed.
Eigenvectors are fundamental concepts in linear algebra. They are special vectors that maintain their direction when a linear transformation is applied - they only get scaled by a factor called the eigenvalue. While regular vectors change both magnitude and direction under transformation, eigenvectors reveal the principal axes along which the transformation acts most simply.