At its core, machine learning processes data. Vectors provide a fundamental mathematical structure for representing individual data points or features. Whether dealing with pixel values in an image, word counts in a document, or measurements from a sensor, representing this information as a vector allows us to apply mathematical operations consistently.

This chapter introduces vectors and their relevance in machine learning contexts. You will learn:

• How data points and features are represented using vectors.
• Basic vector operations: addition, subtraction, and scalar multiplication, along with their geometric meaning.
• Calculating vector magnitude (length) using different norms, such as the $L_1$ (Manhattan) and $L_2$ (Euclidean) norms: $||v||_1 = \sum_i |v_i|$ $||v||_2 = \sqrt{\sum_i v_i^2}$
• Using the dot product to understand vector similarity and projection.
• Measuring distances between vectors in feature space.
• Performing these operations efficiently in Python using the NumPy library.

By the end of this chapter, you will be comfortable manipulating vectors using NumPy and understand how these operations relate to handling data in machine learning tasks.