Linear Transformation Part 2: Foundations, Matrices, and Understanding Linear Transformations, Composition, and Change of Basis with Real-Life Examples with SageMath

Linear Transformations, Composition, and Change of Basis – Simplified + SageMath Code

Welcome Mathsmagic

1. Finding a Linear Transformation Explicitly

🔍 Simple Explanation:

You know how certain "ingredient vectors" (v₁, v₂, v₃) turn into "dishes" (w₁, w₂, w₃). Now, you’re given standard kitchen tools (e₁, e₂, e₃) and asked: "What dishes do these basic tools make?"

Once you know that, you can compute what any input vector produces using a linear combination.

🛠️ Real-Life Use:

Robotics: Moving a robot using known joint motions (v's), and converting new directions (e₁, e₂, e₃) into outputs (w's).

💻 SageMath Code:

2. Composition of Linear Transformations

🔍 Simple Explanation:

Two machines (T then S). The result of first T, then S, is the composition S∘T.

🛠️ Real-Life Use:

Graphics: First transform a 3D object, then project it onto a screen.

💻 SageMath Code:

3. Change of Basis Matrix

🔍 Simple Explanation:

You and your friend describe directions using different landmarks (basis vectors). The change-of-basis matrix lets you translate between those languages.

🛠️ Real-Life Use:

GPS vs. Local Map: Convert from global (lat/lon) to city-grid coordinates.

💻 SageMath Code:

Linear Transformation and Change of Basis

🔍 Simple Explanation:

You apply a transformation T, and want to express it using different coordinate systems (new input/output bases B₁ and B₂). You translate T's matrix A (standard basis) into matrix B (in new bases) using:

\[ B = \rho^2 A \rho^{ - 1} \]

🛠️ Real-Life Use:

Image Processing / FEM: Switching between global and local coordinate descriptions of a system or image.

💻 SageMath Code:

✅ Summary Table

Concept	Analogy	Real Use Case	Key Formula
Linear Transformation	Ingredients → Dishes	Robot control	T(x) = A * x
Composition of Transformations	Two machines in sequence	Graphics pipeline	U = S ∘ T → C = B * A
Change of Basis	Directions in 2 languages	GPS/local maps	P = C⁻¹ * B
Transformation + Change of Basis	Machine in another coordinate frame	Image/wavelet filters	B = ρ₂ * A * ρ₁⁻¹

💬 Call to Action

Did this guide help demystify linear transformations for you? Have any questions or real-life applications you’d like to explore further? Drop a comment below or share your thoughts—I’d love to hear from you!

💡 And if you found this helpful, don’t forget to share it with your friends or colleagues diving into linear algebra!

🔜 Up Next: Eigenvalues and Eigenvectors Part 1

In our next blog, we’ll unlock the powerful concepts of eigenvalues and eigenvectors—essential tools in everything from facial recognition to machine learning! We'll break them down with easy examples, real-world uses, and of course, SageMath code to follow along.

👉 Stay tuned—it’s going to be enlightening!