Linear Transformations

Understanding how data can be manipulated and transformed is crucial in the field of machine learning. Linear transformations provide the mathematical framework for such manipulations, allowing us to translate, rotate, scale, or shear data within vector spaces. In this chapter, you'll gain insights into the nature of linear transformations and how they can be represented using matrices and linear operators.

We'll begin by exploring the concept of linear mappings between vector spaces, delving into the properties that characterize these transformations. You'll learn to express transformations mathematically using matrices, enabling the translation of abstract concepts into tangible computations. The chapter will also uncover the significance of the kernel and image of a transformation, highlighting their roles in understanding the effects of transformations on vector spaces.

Additionally, we'll connect linear transformations to practical applications in machine learning. By the end of this chapter, you will have a solid grasp of how linear transformations can simplify complex data manipulations and enhance the efficiency of algorithms. This knowledge will serve as a foundation for more advanced topics, such as eigenvectors and eigenvalues, which are crucial for understanding dimensionality reduction and other key machine learning techniques.