AI is inevitable, and I've come to embrace that reality. With this in mind, I’m preparing to become an AI engineer.
To master AI, one of the foundational concepts you need to understand is Linear Algebra. In this article, I’ll dive into vectors, a key building block of linear algebra, which I’ve been studying as part of my learning journey through Khan Academy. These notes are my way of consolidating and sharing the essential concepts I’ve encountered along the way.
Vector = Magnitude + Direction
A vector is defined by both its magnitude (size) and its direction.
Example 1: "A car is moving at 5 km/h"
Is this a vector?
No. While 5 km/h represents the magnitude, there's no direction specified. Without direction, it's not a vector.
Example 2: "A car is moving at 5 km/h east"
This is a vector.
Why? Because it includes both a magnitude (5 km/h) and a direction (east). A vector requires both components to be meaningful.
How to Represent Vectors
- 2D Representation:
One common way to represent vectors in two dimensions is by using coordinates, like (5, 0), where 5 represents the magnitude along the x-axis, and 0 indicates no movement along the y-axis.
- Alternate Representation (Magnitude Only):
In practice, vectors are often simplified and represented as [5], where the magnitude is shown without explicitly writing out the direction. This is commonly used in certain contexts, like velocity or speed.
- 3D and Beyond:
In three dimensions, we can represent vectors with three components, e.g., (5, 0, 3), which would indicate a vector with 5 units along the x-axis, no movement along the y-axis, and 3 units along the z-axis.
Beyond three dimensions, representation becomes more abstract, though it's still possible using vector notation and mathematical frameworks.
Example of 2D & 3D vector
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