Math for Machine Learning - Chapter 8: Linear Systems of Equations

This quiz delves into advanced aspects of systems of linear equations, a cornerstone of mathematics for machine learning. Moving beyond basic solution methods, these questions rigorously test your understanding of the underlying theory, geometric interpretations, and the conditions for solution existence and uniqueness. We explore the profound differences between the 'row picture' (viewing equations as intersecting hyperplanes) and the 'column picture' (expressing the right-hand side vector as a linear combination of the matrix's column vectors). Key concepts include the rank of a matrix ($rank(A)$), the column space ($C(A)$), the null space ($N(A)$), and their critical roles in determining system consistency ($b \in C(A)$ implies consistency) and uniqueness ($N(A)$ containing only the zero vector implies uniqueness). You will apply these concepts to various matrix dimensions ($m \times n$) and scenarios, including overdetermined and underdetermined systems, homogeneous systems ($A\mathbf{x}=\mathbf{0}$), and the general solution structure $\mathbf{x} = \mathbf{x}_p + \mathbf{x}_h$. Prepare to analyze the implications of matrix properties like invertibility, linear dependence of rows/columns, and the powerful rank-nullity theorem. This quiz is designed to solidify your expertise in deciphering the intricate behavior of linear systems, essential for fields like regression, optimization, and dimensionality reduction in machine learning.

Key Formulas:

$A\mathbf{x} = \mathbf{b}$

$\text{Consistency condition: } rank(A) = rank([A|\mathbf{b}])$

$\text{Rank-Nullity Theorem: } rank(A) + dim(N(A)) = n \quad (\text{where } n \text{ is the number of columns of } A)$

$\text{General Solution: } \mathbf{x} = \mathbf{x}_p + \mathbf{x}_h, \quad \text{ where } A\mathbf{x}_p = \mathbf{b} \text{ (particular solution) and } A\mathbf{x}_h = \mathbf{0} \text{ (homogeneous solution)}$