Module 1: AI Foundations

I. Why Math Matters for AI?

Artificial Intelligence is not magic; it's advanced applied mathematics. To understand how models "learn", we need to understand the tools they use to measure error and optimize parameters.

The Big Picture Neural Networks are just layers of composite functions aligned with linear algebra operations, optimized via calculus (gradients) and statistical reasoning (loss functions).

Calculus

Used to find the "slope" of errors, telling the model which direction to move to improve.

Linear Algebra

Used to process massive amounts of data inputs simultaneously (parallelism).

II. Functions: The Building Blocks

A function is a relationship that maps an input to an output. In AI, neurons are functions.

Linear Function

$$ f(x) = mx + b $$

Slope (m)

1.0

Intercept (b)

0.0

Quadratic Function

$$ f(x) = a(x-b)^2 $$

Scale (a)

1.0

Shift (b)

0.0

III. Derivatives: Measuring Change

The derivative tells us the rate of change (slope) at any specific point on a curve. This slope is the key signal used to train neural networks.

Derivative Simulator

Visualize the tangent line (red) for $ f(x) = x^2 $

Point (x)

2.0

IV. Vectors: Data Containers

Vectors are ordered lists of numbers describing a point in space. In AI, they represent features (e.g., [Age, Income, Debt]).

Vector Operations

Explore Magnitude and Normalization

X Component

3.0

Y Component

4.0

Magnitude: 5.00

Normalized: [0.60, 0.80]

V. Gradient Descent

Gradient Descent is the algorithm used to train models. It iteratively moves towards the lowest error (loss) by following the negative gradient.

Step-by-Step Descent

Optimization of $ L(\theta) = (\theta-3)^2 $

Learning Rate ($ \alpha $)

0.10

                                    Iteration: 0

                                    Position: θ = 4.0

                                    Loss: L = 1.0

VI. Probability & Uncertainty

AI models deal with uncertainty. Probability distributions allow us to quantify likelihoods.

Normal Distribution (Gaussian)

$$ P(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

Mean (μ)

0.0

Std Dev (σ)

1.0

Bayes' Theorem $$ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} $$ The foundation of probabilistic reasoning, allowing us to update beliefs based on new evidence.