Mathematics for Machine Learning - Chapter 1: Unveiling Vectors Quiz

This quiz assesses your understanding of fundamental vector concepts as introduced in Chapter 1 of 'Mathematics for Machine Learning'. It covers vectors from physics, computer science, and pure mathematics perspectives, their foundational operations (addition and scalar multiplication), the concept of linearity, and vector dimension. Mastery of these concepts is crucial for building a strong foundation in linear algebra and machine learning.

Key Formulas/Concepts Covered:

• Vector Addition: $\mathbf{v} + \mathbf{w} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix} + \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{pmatrix} = \begin{pmatrix} v_1+w_1 \\ v_2+w_2 \\ \vdots \\ v_n+w_n \end{pmatrix}$

• Scalar Multiplication: $c \mathbf{v} = c \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix} = \begin{pmatrix} c v_1 \\ c v_2 \\ \vdots \\ c v_n \end{pmatrix}$

• Linearity Properties:

  1. Additivity: $f(\mathbf{v} + \mathbf{w}) = f(\mathbf{v}) + f(\mathbf{w})$

  2. Homogeneity of Degree 1: $f(c \mathbf{v}) = c f(\mathbf{v})$