The single most important object in all of machine learning and quantitative finance.
What’s the single most important object in all of machine learning and quantitative finance? It’s not a neural network or a fancy algorithm. It’s something much simpler: the vector. But what is a vector?
If you ask a physicist, a programmer, and a data scientist, you might get three different answers. The key to mastering linear algebra is understanding that they're all talking about the same powerful idea from different perspectives. Let's break down the two most important views.
To a physicist, a vector is an arrow floating in space. It's an object defined by two things and two things only:
Think of concepts like velocity, force, or acceleration. A wind blowing at 15 mph to the northeast is a vector. It doesn't matter if you're measuring that wind in London or Tokyo; as long as the speed (length) and direction are the same, it's the same vector.
To a data scientist or a programmer, a vector isn't an arrow; it's an ordered list of numbers. That's it. Seriously.
This house is a vector:
The entire foundation of applied linear algebra rests on one, beautiful idea: these two views are just different languages for the same concept.
We can translate from the "list of numbers" to the "arrow" by creating a coordinate system. To represent the vector , we simply:
This act of visualizing a list of numbers as a single point (or arrow) in space is the most important skill you will learn.
Every single data point can be thought of as a vector in a high-dimensional space.
Why is this so powerful? Because once your data points become vectors, you can start asking geometric questions.
This translation is the key. We take our messy, real-world data (lists of numbers) and place it into a clean, geometric world (vector spaces) where we can use the powerful tools of linear algebra to find patterns.