What are Vectors? A Complete Guide to Understanding Linear Algebra's Foundation

Linear algebra is built upon a single, deceptive foundation: the vector. But asking what a vector is will yield three different answers depending on who you ask. The physics student sees an arrow pointing through space with a specific length and direction.

In contrast, the computer science student views the vector as an ordered list of numbers. To them, a house might be a two-dimensional vector where the coordinates represent square footage and price. The order of these numbers is the only thing that matters.

However, the mathematician seeks a higher level of abstraction that generalizes both perspectives. They argue that a vector can be anything, provided there is a sensible way to add them and multiply them by numbers.

Therefore, we must master the interplay between these three distinct lenses to truly understand the field. We begin with the geometric view to build intuition, then translate it into the numerical view to perform calculations.

The vector is the fundamental building block that unites geometry and data.

For our purposes, we will treat vectors as arrows anchored at the origin of a coordinate system. This convention allows us to map geometric intuition directly onto numerical data without the vector floating away. The origin serves as the center of the universe and the starting point for every vector we draw.

In fact, the first number tells you how far to move along the horizontal x-axis. Positive numbers indicate rightward motion, while negative numbers indicate a shift to the left. The second number dictates the vertical movement parallel to the y-axis.