What is a Vector? Translating the Real World into Code

Feature RepresentationVectors and DimensionsBuilding a Custom Vector Class

🧠The Theory

AI/ML Concept: Feature Representation

Computers do not natively understand real-world concepts like a "house," an "image," or a "song." To feed data into a Machine Learning model, we must translate these entities into vectors. This critical first step is called feature representation.

Tabular Data: If we are building a model to predict house prices, we might define a house using three features: number of bedrooms, number of bathrooms, and age in years. A 3-bedroom, 2-bathroom house built 15 years ago becomes a data point in 3D space: $\vec{h} = [3, 2, 15]$ .
Image Data: A grayscale image is represented as a vector where each element corresponds to the brightness of a single pixel.
Text Data: Words are mapped to high-dimensional vectors (often 300+ dimensions) where the numbers represent semantic meaning.

Every object an AI ever interacts with is converted into this numerical format so the underlying mathematical engine can process it.

📐The Math

What is a Vector?

In mathematics, a vector is an ordered list of numbers that represents a point or a direction in a coordinate space. The number of elements in the vector determines its dimension ( $n$ ).

A 2-dimensional vector represents a point on a flat plane (x, y axes):
$\vec{v} = [x_1, x_2]$
A 3-dimensional vector represents a point in physical space (x, y, z axes):
$\vec{v} = [x_1, x_2, x_3]$

While humans struggle to visualize anything beyond 3 dimensions, the mathematical rules remain exactly the same for an $n$ -dimensional space, allowing us to represent high-dimensional vectors algebraically:
$\vec{v} = [x_1, x_2, \dots, x_n]$

⚙️The Code

class Vector:
    def __init__(self, attributes: list[float]):
        self.attributes = attributes
    
    def __sub__(self, other: "Vector") -> "Vector":
        if isinstance(other, Vector):
            if len(self.attributes) != len(other.attributes):
                raise ValueError("Vectors must have the same dimension for subtraction.")
            return Vector([s - o for s, o in zip(self.attributes, other.attributes)])
        else:
            raise TypeError("Unsupported operand type for -: 'Vector' and '{}'".format(type(other).__name__))
        
    def __repr__(self) -> str:
        attributes_str = ", ".join("{:.2f}".format(a) for a in self.attributes)
        return "Vector({})".format(attributes_str)


# Example Usage: Simulating two houses with 4 features (bedrooms, bathrooms, age, sqft)
house_A = Vector([3, 2, 15, 2000]) 
house_B = Vector([4, 3, 10, 2500])

# Computes the difference in features: [-1.00, -1.00, 5.00, -500.00]
difference = house_A - house_B 
print(f"Feature Difference: {difference}")

Code Breakdown

class Vector: We define a custom class to represent our mathematical vector, bypassing NumPy initially to understand the raw mechanics.
def __init__(self, attributes): Initializes the vector with an arbitrary sequence of numbers. Unlike a hardcoded 3D vector (x, y, z), this allows for $n$ -dimensional feature representation.
if isinstance(other, Vector): Guards against invalid operations. We can only subtract a Vector from another Vector.
if len(self.attributes) != len(other.attributes): Dimensionality Check: Mathematically, vector addition and subtraction are only defined for vectors residing in the exact same dimensional space.
zip(self.attributes, other.attributes): Pairs corresponding elements $(a_i, b_i)$ from both vectors together.
[s - o for s, o in ...]: List Comprehension: Iterates through the zipped pairs, performs the element-wise subtraction $a_i - b_i$ , and builds the new resulting list in a single, optimized pass.
return Vector(...): Wraps the resulting list back into a new Vector object, allowing for method chaining later on.

Euclidean Distance: Measuring Similarity