Syllabus Map

• Study map: Syllabus Study Map

Core Idea

• A linear system is written as $Ax=b$.
• Rank tells how many independent constraints/features exist.

Key Facts

• Unique solution: full rank square system.
• Infinite solutions: at least one free variable.
• No solution: inconsistent equations.

Practical Notes

Rank deficiency implies redundant information.

• Correlated features can reduce effective dimensionality.

REF/RREF provide structural diagnostics.

• Use pivots to identify dependencies and solvability.

Why This Matters for ML

• Least-squares fitting and many closed-form estimators reduce to solving linear systems.
• Rank reveals multicollinearity and redundant features, which affect model identifiability.
• Underdetermined systems motivate regularisation and minimum-norm solutions.
• Pivot structure from REF/RREF gives a practical diagnostic for solvability and feature dependence.