Linear Algebra & Calculus Refresher

Introduction

Linear algebra and calculus form the mathematical foundation of machine learning. Understanding these concepts is crucial for grasping how algorithms learn from data.

Key Concepts

Vectors and Matrices

Vectors represent data points, while matrices represent transformations or collections of data. Matrix operations are fundamental to neural network computations.

Derivatives and Gradients

Derivatives measure how functions change. In ML, gradients (multi-dimensional derivatives) tell us how to adjust parameters to minimize loss.

Chain Rule

The chain rule allows us to compute derivatives of composite functions, which is essential for backpropagation in neural networks.

Interactive Matrix Multiplication

Adjust matrix dimensions and values to see how matrix multiplication works. The result updates automatically!

Matrix A Dimensions

×

Matrix B Dimensions

2
×

B rows must equal A columns

Matrix A

2 × 2
×

Matrix B

2 × 2
=

Result

4
4
10
8
2 × 2

Calculation Steps:

Result[0,0] = (1 × 2) + (2 × 1) = 4
Result[0,1] = (1 × 0) + (2 × 2) = 4
Result[1,0] = (3 × 2) + (4 × 1) = 10
Result[1,1] = (3 × 0) + (4 × 2) = 8

Interactive Derivative Visualization

See how derivatives represent the rate of change of functions. The derivative tells us the slope at any point.

Function
Derivative

Select Function

Key Insights:

  • The derivative is positive when the function is increasing
  • The derivative is negative when the function is decreasing
  • The derivative is zero at local maxima and minima
  • The magnitude shows how fast the function is changing

Why This Matters for ML

Data Representation: Input data is typically represented as vectors or matrices. Images are 2D/3D matrices, text is encoded as vectors.

Model Parameters: Neural network weights are matrices that transform inputs through layers.

Optimization: Calculus provides the tools (gradients) to find the best parameters by minimizing loss functions.

Backpropagation: The chain rule enables efficient computation of gradients through complex neural networks.