EE-869 Linear Systems Theory – Maths Matrix Refresher & Cheat Sheet, Fall Semester 2025, Comprehensive Lecture Summary

EE-869 Linear Systems Theory​
FALL SEMESTER 2025
Ghulam Mustafa




December 2025

Linear Systems Theory: Math Refresher &
Cheat Sheet
A Layman’s Guide to the Concepts (Up to Page 37)

1. The Vocabulary of "Space" (The Rules of the Game)
Before we solve equations, we must define where they live.

●​ Field 𝐅: The "Palette of Numbers."
○​ R (Reals): Physical quantities (mass, voltage).
○​ ℂ (Complex): Oscillations and frequencies.
●​ Vector Space: The "Playground." A set where you can add vectors and scale
them without leaving the set.
●​ Span: The "Reach." The set of all possible points you can reach using a specific
list of vectors.
○​ Visual: The span of two arrows on a desk is the entire flat surface of the
desk.
●​ Range (Image): The "Output Possibilities."
○​ For a system matrix A, the Range is the set of all possible outputs it can
produce.
○​ Significance: If a state is outside the Range, the system cannot go there.
●​ Orthogonality: "Independence."
○​ Two vectors are orthogonal if their inner product is zero (<x, y> = 0).
○​ Significance: Allows us to separate signals (decoupling) and filter noise
cleanly.




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