Math for Machine Learning - Chapter 4 :Dot Product and Geometric Interpretations

This expert-level quiz delves into the dot product, a fundamental operation bridging algebra and geometry in machine learning. It is designed to test your deep understanding of its properties, geometric interpretations, and applications. Questions will challenge your ability to work with projections, orthogonality, angles between high-dimensional vectors, and the connection between the dot product and concepts like cosine similarity and hyperplanes. Mastery of this topic is crucial for understanding algorithms ranging from Support Vector Machines (SVM) to neural network operations and recommendation systems.

Key Formulas:

• Algebraic Dot Product: $\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = \mathbf{a}^T \mathbf{b}$

• Geometric Dot Product: $\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos(\theta)$

• Vector Projection of $\mathbf{b}$ onto $\mathbf{a}$: $\text{proj}_{\mathbf{a}}(\mathbf{b}) = \left( \frac{\mathbf{b} \cdot \mathbf{a}}{\|\mathbf{a}\|^2} \right) \mathbf{a}$

• Scalar Projection of $\mathbf{b}$ onto $\mathbf{a}$: $\text{comp}_{\mathbf{a}}(\mathbf{b}) = \frac{\mathbf{b} \cdot \mathbf{a}}{\|\mathbf{a}\|}$

• Cosine Similarity: $\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}$

• Orthogonality Condition: $\mathbf{a} \cdot \mathbf{b} = 0$