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.
1
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.
1
