Introduction to Linear Algebra

Basic-Mathematics tutorial · PySpark.in

Understanding the Importance of Linear Algebra in Data Science

• Data Representation (Vectors & Matrices)

• Machine Learning Algorithms

• Dimensionality Reduction (PCA)

• Optimization (Gradient Descent)

• Data Manipulation and Transformations

• Deeper Understanding and Intuition

Core Concepts of Linear Algebra

Scalars, Vectors, and Matrices

• Scalar: A single number (e.g., the number of rooms in a house).

• Vector: A list of numbers, often representing a single data point or a single feature (e.g., [1200, 2], representing area and number of rooms).

• Matrix: A grid of numbers, where each row can be a data point and each column a feature. Your entire dataset is a matrix.

Mathematical Operations

• Matrix addition:



• Matrix subtraction:

• Matrix multiplication (Crucial for model training):

Linear Transformations

• Functions that change vectors and vector spaces in a predictable way.

• Example: scaling or rotating images.

Eigenvalues and Eigenvectors

• Used in advanced techniques like reducing the complexity of data.

Applications of Linear Algebra

• Data Representation and Manipulation

• Machine Learning and Artificial Intelligence

• Computer Graphics and Vision

• Optimization

Scalars

A scalar is a single numerical value that represents a magnitude or quantity and has no direction.

Example:

• Car speed: A car traveling at 60 km/h → The number 60 is the scalar value.

Vectors

A vector is a numerical value that has both magnitude and direction.

Example: Student Data
Imagine a dataset with information about students:

• IQ score

• Study hours

• Pass/Fail status

Vector Operations

• 2(a) Vector Addition:



• 2(b) Vector Subtraction:



• 2(c) Vector Multiplication:

 Vector Dot Product:






 Vector Cross Product:

Matrices in Data Science

Uses

• Data Representation

• Image Representation

• Confusion Matrix

Matrix Operations in Machine Learning Algorithms

• Matrix Addition

• Matrix Subtraction

• Matrix Multiplication

Some Important Properties

• Properties of Vectors

• Properties of Matrices

Calculating the Eigenvalue and Eigenvector: used for PCA:

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