Linear Algebra

What is Linear Algebra? #

Linear Algebra is a branch of mathematics that deals with vectors, matrices, and their operations. It is widely used in Data Science and Machine Learning.

Vectors #

A vector is a list of numbers representing magnitude and direction.

Example: #

v = [2, 4, 6]

Vector Operations #

Addition:

[1, 2] + [3, 4] = [4, 6]

Scalar Multiplication:

2 × [1, 2] = [2, 4]

Matrices #

A matrix is a 2D array of numbers (rows & columns).

Example: #

A = [[1, 2],
     [3, 4]]

Matrix Operations #

1. Addition #

(Add corresponding elements)

A + B = [[1+5, 2+6],
         [3+7, 4+8]]

Multiplication

A × B = [[(1×5 + 2×7), (1×6 + 2×8)],
         [(3×5 + 4×7), (3×6 + 4×8)]]

Transpose #

(Convert rows into columns)

Aᵀ = [[1, 3],
      [2, 4]]

Basic Python Example

import numpy as np

# Vectors
v1 = np.array([1, 2])
v2 = np.array([3, 4])

print("Addition:", v1 + v2)
print("Scalar:", 2 * v1)

# Matrices
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

print("Matrix Add:\n", A + B)
print("Matrix Multiply:\n", A.dot(B))
print("Transpose:\n", A.T)

Importance in Data Science #

• Used in machine learning algorithms
• Helps in handling large datasets
• Important for deep learning and AI