Linear Algebra
You sit at your desk, elbows planted, hands cradling your face as you stare into the silent brilliance of the screen before you.
Not a screen in the old sense—a floating pane of smartglass, perfectly transparent but alive with light. Lines of code shimmer across its surface, radiant in neon hues. They appear suspended inside the glass, like strands of DNA embedded in crystal. It feels as if you could reach forward, tear them from their logic-bound prison, and reshape them with your bare hands.
The office around you hums with elegant sterility. It’s the size of a small apartment—generous, for KadoMin standards—and tastefully austere. White walls absorb light with the softness of snow, while crimson accents—rugs, lighting strips, a vase of synthetic roses—offer just enough contrast to feel deliberate. Everything is minimalist. Perfect. Cold.
Except the chair. Or rather, the throne. High-backed, armored in leather and alloy, it supports you like a monarch at war with unseen enemies.
Your eyes blur. The screen begins to melt into the background—abstract art, clean lines, indistinguishable from the code. Thoughts spiral.
Why does this only happen to some of the Sentinels?
Not a production fault—you ruled that out days ago. The anomalies are global, scattered across units from different lines, facilities, even continents. No shared serials, no overlapping firmware updates.
You swipe your hand through the air.
The display obeys. The code vanishes. In its place: a police report. Centered is the face of a Sentinel—porcelain skin, symmetrical features, eyes too wide, too empty. A line of static cuts through the image like a scar.
You skim the text. Your voice breaks into a whisper.
“Nothing. Nothing… still nothing…”
Until you reach the incident log.
Incident Report – February 17, 2049 – 03:13 AM
Caller: Female, early 30s, location: Wexler District, Neo-Paris
“It’s just standing there!”
“Ma’am, please stay calm. What’s standing there?”
“My Sentinel. It’s not supposed to be here—it’s not supposed to be here!”
“How did it enter your residence?”
“I don’t know! It’s staring at me. Not moving. Just watching. I locked it in the garage last night. I didn’t even call it. Please send someone, please—“
You exhale, slow and bitter.
“This one’s even worse than the last…”
Most deviations have been minor—slight hesitations in command execution, forgotten items, subtle mismatches in behavior. An uncanny glance. A pause too long. Harmless, on the surface.
But this—this was different. The unit broke containment. Uninvited. No command. No trigger. And still—no pattern.
No common model, no shared neural template, no matching age, no firmware drift. Nothing. You’ve been here for weeks, drowning in data, theories, and silence. Outside these walls, the world is starting to notice. Headlines swirl. Whispers of recalls. Demands for answers. KadoMin is under fire, and you are their shield. Your mind spins, looping through dead-end hypotheses, then discarding them seconds later.
Loop after loop. Spin after spin. Until—
Knock.
You jolt. A single, deliberate rap on the office door. You straighten in your chair, adjust your collar, and speak aloud, voice steady despite the storm within.
“Come in.”
How can we hope to fix or improve something if we don’t truly understand how it works?
Applying machine learning is easy. With just a few lines of code, you can load a library, choose a pre-built algorithm, feed it your data, and watch it work. You don’t need to be a math expert to use these tools.
But using machine learning isn’t our end goal—we’re aiming for true mastery. We want to know how and why it works under the hood. And to do that, we need a solid foundation in math. That’s exactly what this course is here to provide.
In this course, we’ll focus on building that foundation through one of the two core mathematical pillars of machine learning: Linear Algebra. (The other major pillar is Probability Theory, which we’ll cover in a seperate course.)
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating them—something you’ve likely encountered in school. If you’ve ever solved equations, worked with inequalities, factored expressions, or tackled quadratic equations, you’ve done algebra. You may have also solved systems of equations with multiple unknowns.
Simple Equation:
\( x + 20 = 10 \)
Simple Inequality:
\( x + 20 \geq 10 \)
System of Linear Equations:
\( \begin{cases}
2x + y = 10 \\
x \; – \; y = 2
\end{cases} \)
Linear Algebra is a specific area within algebra that focuses on vectors and the rules for manipulating them. Understanding vectors—and the intuition behind them—is key to grasping many machine learning concepts.
This course is inspired by several excellent (and free) resources, including:
- Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong
- Deep Learning by Ian Goodfellow, Yoshua Bengio, and Aaron Courville
- The YouTube videos and website of 3Blue1Brown, whose visual explanations make linear algebra remarkably intuitive
I highly recommend watching 3Blue1Brown’s videos on Linear Algebra (Essence of Linear Algebra) as you go through this course. His visual style brings the concepts to life and will give you a strong intuitive grasp of what linear algebra is really about. Watching these alongside the course content is the ideal way to learn.
“Wait—you mentioned vectors. But what are vectors, exactly?”
I’m glad you asked!