I’ve gone through your uploaded handwritten notes (the “Contractor Kartavya’s Conundrum / Ekalavya’s Excursions” material) and distilled them into two clean outputs:
Structured study notes (crisp, exam/learning-friendly)
A conceptual blog post (story-driven, intuitive, readable)
Both are derived directly from the content in the PDF, especially pages 1–5 and the later linear algebra sections on functions, matrices, inverses, and transformations .
1. Clean Notes (Structured)
Theme 1: Contractor Kartavya’s Conundrum
Problem statement
Kartavya wants to determine the optimal statewise distribution of workers to maximize productivity.
Parameters:
( k = 120 ) months
( n = 30 ) states
Productivity over time is modeled using a system of linear equations:
[
\begin{aligned}
a_{11}s_1 + a_{12}s_2 + \dots + a_{1n}s_n &= P_1 \
a_{21}s_1 + a_{22}s_2 + \dots + a_{2n}s_n &= P_2 \
\vdots \
a_{k1}s_1 + a_{k2}s_2 + \dots + a_{kn}s_n &= P_k
\end{aligned}
]
Key idea
This is a system-level optimization problem
Best strategy: Outsource decision-making to a “game” or algorithmic process
Introduces the idea of learning via error minimization
Theme 2: Outsourcing as Optimization
How Kartavya outsources
Defines an error function
Uses an iterative method metaphorically called
“Blind man’s walk down the hill”Intuition for gradient descent
Move step-by-step in the direction that reduces error
Insight
Even without deep math, arithmetic + iteration can converge to good solutions
Theme 3: Functions and Inverses (Ekalavya’s Excursions)
Scalar Functions
Example:
[
f(x) = x^2 \Rightarrow f^{-1}(y) = \sqrt{y}
]Understanding a function deeply allows construction of its inverse
Theme 4: Matrices as Functions
A matrix behaves like a function mapping vectors to vectors.
Example:
[
f:
\begin{pmatrix}
x \ y
\end{pmatrix}
\mapsto
\begin{pmatrix}
2x + y \
x + 3y
\end{pmatrix}
]
This is a function ( f: \mathbb{R}^2 \to \mathbb{R}^2 )
Question: Is it invertible?
Theme 5: Finding the Inverse (Hook or Crook)
Solve:
[
\begin{cases}
2\alpha + \beta = a \
\alpha + 3\beta = b
\end{cases}
]
Solution:
[
\begin{pmatrix}
\alpha \
\beta
\end{pmatrix}
\begin{pmatrix}
\frac{3a-b}{5} \
\frac{2b-a}{5}
\end{pmatrix}
]
Matrix inverse:
[
\begin{pmatrix}
2 & 1 \
1 & 3
\end{pmatrix}^{-1}
\begin{pmatrix}
\frac{3}{5} & -\frac{1}{5} \
-\frac{1}{5} & \frac{2}{5}
\end{pmatrix}
]
Theme 6: Linear Combinations & Span
Any point in the plane can be written as:
[
\alpha u + \beta v
]If vectors (u, v) are linearly independent → mapping is 1–1 and onto
Theme 7: Composition of Linear Maps
If:
[
f(x,y) = (2x+y,\ x+3y)
]
[
g(a,b) = (a+b,\ 4a+5b)
]
Then:
[
g(f(x,y)) = (3x+4y,\ 13x+19y)
]
➡️ Composition corresponds to matrix multiplication
Theme 8: Geometric Transformations
Rotation by 90° clockwise:
[
\begin{pmatrix}
0 & 1 \
-1 & 0
\end{pmatrix}
]Reflections and rotations are also linear transformations
Geometry ↔ algebra via matrices
2. Blog Post (Conceptual & Story-Driven)
From Blind Walks to Brilliant Solutions: How Linear Algebra Explains Learning
Kartavya isn’t a mathematician. He’s a contractor with a problem.
He has workers scattered across 30 states and productivity data spanning 120 months. The question haunting him is deceptively simple:
How should I distribute my workers so productivity is maximized?
Behind the scenes, this is a massive system of equations—far too complex to solve by hand. So Kartavya does something radical.
He outsources thinking itself.
Learning Without Knowing the Math
Kartavya defines an error—how far reality is from what he wants. Then he lets the town try random steps, keeping only those that reduce the error.
It’s like a blindfolded person walking downhill, step by step.
You already know this idea by another name: gradient descent.
Learning, it turns out, doesn’t require genius—just feedback and direction.
Ekalavya’s Real Lesson
Ekalavya asks a deeper question: What does it mean to reverse a process?
If squaring turns 3 into 9, what turns 9 back into 3?
This curiosity leads us from simple functions to matrices—objects that don’t just compute numbers, but move space itself.
Matrices Are Functions in Disguise
A matrix isn’t just numbers in a box. It’s a function:
It takes a point
Moves it
Stretches it
Rotates it
Or flips it
And sometimes—if you’re lucky—you can undo it perfectly.
That’s what an inverse really is:
A guarantee that no information was lost.
Why This Matters
Machine learning is just Kartavya’s outsourcing, scaled up
Neural networks are stacked matrix functions
Invertibility is about reversibility and trust
Geometry, algebra, and optimization are the same story told differently
Final Thought
Math isn’t about symbols.
It’s about control:
Can you predict?
Can you reverse?
Can you improve step by step?
Kartavya didn’t learn linear algebra.
Linear algebra learned Kartavya.
If you want, I can:
Turn this into LaTeX notes
Simplify it for high-school / JEE / undergrad
Convert the blog into Medium / Substack style
Add visual explanations or diagrams
Just tell me how you want to use it.
