Vectors as Data Points

Linear Algebra for Machine Learning · Axiom Academy

Representing information as ordered lists of numbers A vector is an ordered collection of numbers, arranged in a sequence. Think of it as a list where position matters. A vector in ℝⁿ is an ordered tuple of n real numbers written as: The numbers are called components or coordinates of the vector. Each position has meaning! In a 2D vector [3, 4], the first component is 3 and the second is 4. They're not interchangeable — [4, 3] is a different vector! In machine learning and data science, we represent real-world objects as vectors. Each component represents a measured property or feature. [Grade, Test Score, Attendance Rate] Each object becomes a point in high-dimensional space. This lets us use mathematics to find patterns, similarities, and relationships in data. When we represent multiple data points as vectors, they all live in the same space — called the feature space . The n-dimensional space ℝⁿ where each dimension represents a feature. Each data point is a position in this space. Example: If we have 3 features, our data points live in 3D space. The distance between these points in feature space tells us how similar the houses are. Close points are similar; far points are different. Real data often has many features, creating very high-dimensional vectors. Image: A 28×28 pixel image = 784-dimensional vector (one value per pixel) Document: Word frequencies for 10,000 words = 10,000-dimensional vector Gene Expression: Activity of 20,000 genes = 20,000-dimensional vector

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