Understanding Vectors: The Building Blocks of Artificial Intelligence

The Concept of Vectors

Vectors are one of the core aspects of Artificial Intelligence. To understand the mathematics behind concepts like artificial intelligence, machine learning, or deep learning, one must start with the basics—Vectors.

What are Vectors?

In Linear Algebra, vectors are mathematical objects that have both direction and magnitude. If a quantity has only magnitude and no direction, it is called a scalar.

• Magnitude: The distance from the origin (0,0) to the given point (x, y).

In Computer Science, vectors are often represented as ordered lists. For example, in Python, a single list of elements like [2, 5] represents a 2D vector. The number of elements in the list determines the dimension of the vector.

• If a list contains three elements (e.g., [3, 6, 7]), it represents a 3D vector, meaning it will be plotted in a 3D space with x, y, and z axes.

We can calculate the magnitude of a vector using the following formula:

∣v∣=x2+y2|v| = \sqrt{x^2 + y^2}

In Python, the NumPy library provides a built-in method to calculate the magnitude of a vector:

import numpy as np
vector = np.array([3, 4])
magnitude = np.linalg.norm(vector)
print(magnitude)  # Output: 5.0
        

Types of Vectors

There are various types of vectors, including:

• Zero Vector: A vector where all elements are zero.
• Unit Vector: A vector with a magnitude of 1.
• Equal Vectors: Two vectors with the same direction and magnitude.
• Negative Vectors: Two vectors with the same magnitude but opposite directions.
• Parallel Vectors: Vectors with the same angle but different magnitudes. The direction can be same or opposite.
• Orthogonal Vectors: Vectors that are perpendicular to each other.

Vector Operations

You can perform arithmetic operations between two vectors, but they must satisfy one condition: ➡ Both vectors should have the same dimensions.

Vector Addition

When adding two vectors of the same dimension, you add their corresponding elements:

v1 = np.array([1, 2, 3])
v2 = np.array([4, 5, 6])

v_add = v1 + v2
print(v_add)  # Output: [5, 7, 9]
        

The same logic applies to vector subtraction and element-wise multiplication.

Dot Product (Scalar Product)

The dot product is different from element-wise multiplication. Instead of returning a vector, it produces a scalar value.

v_dot = [2, 3, 6] . [7, 1, 5]

v_dot = [2*7 + 3*1 + 6*5]

v_dot = [47]

Python Example:

v1 = np.array([2, 3, 6])
v2 = np.array([7, 1, 5])

dot_product = np.dot(v1, v2)
print(dot_product)  # Output: 47
        

Since the dot product results in a scalar, it has magnitude but no direction.

Vector-Scalar Multiplication

This operation is straightforward—each element of the vector is multiplied by the scalar:

scalar = 5
vector = np.array([1, 2, 3])

result = scalar * vector
print(result)  # Output: [5, 10, 15]
        

Conclusion

Vectors are the foundation of many advanced AI and machine learning concepts, from neural networks to computer vision. Mastering vectors and their operations will help you develop a deeper understanding of mathematical models in AI.

🚀 If you're diving into AI, start with vectors—the fundamental building blocks of intelligent systems!