Skip to content

Day 38: Math Foundations - Linear Algebra

Overview

Linear algebra underpins much of machine learning. This lesson revisits the building blocks—vectors, matrices, and eigendecomposition—so you can reason about data transformations and model behaviour with confidence.

Key Concepts

  • Vectors: Mathematical objects with magnitude and direction that often represent individual data points or feature sets.
  • Matrices: Rectangular arrays that hold datasets and enable transformations when combined with vectors or other matrices.
  • Dot Product: A measure of similarity between two vectors computed by multiplying corresponding elements and summing the results.
  • Eigenvectors and Eigenvalues: Special vector–scalar pairs that describe how a matrix stretches space; essential for dimensionality reduction techniques such as PCA.

Practice Exercises

  1. Vector Operations: Use vector_operations() to compute the sum, difference, and dot product for [1, 2, 3] and [4, 5, 6].
  2. Matrix Operations: Call matrix_operations() to produce the sum and product for the matrices [[1, 2], [3, 4]] and [[5, 6], [7, 8]].
  3. Eigen-analysis: Explore the eigenvalues and eigenvectors of [[4, 1], [2, 3]] via eigen_analysis().

How to Use This Folder

  • Run the worked examples: python Day_38_Linear_Algebra/solutions.py
  • Execute the automated checks: pytest tests/test_day_38.py

Previous: Day 37 – Day 37: Conclusion & Your Journey Forward • Next: Day 39 – Day 39: Math Foundations - Calculus

You are on lesson 38 of 108.

Additional Materials

solutions.py

View on GitHub

solutions.py
import numpy as np


def vector_operations(v1=None, v2=None):
    """Return the sum, difference, and dot product of two vectors."""

    v1_array = np.array([1, 2, 3]) if v1 is None else np.asarray(v1)
    v2_array = np.array([4, 5, 6]) if v2 is None else np.asarray(v2)

    vector_sum = v1_array + v2_array
    vector_difference = v1_array - v2_array
    dot_product = np.dot(v1_array, v2_array)

    return vector_sum, vector_difference, dot_product


def matrix_operations(m1=None, m2=None):
    """Return the sum and product of two matrices."""

    m1_array = np.array([[1, 2], [3, 4]]) if m1 is None else np.asarray(m1)
    m2_array = np.array([[5, 6], [7, 8]]) if m2 is None else np.asarray(m2)

    matrix_sum = m1_array + m2_array
    matrix_product = np.dot(m1_array, m2_array)

    return matrix_sum, matrix_product


def eigen_analysis(matrix=None):
    """Return the eigenvalues and eigenvectors for the provided matrix."""

    matrix_array = np.array([[4, 1], [2, 3]]) if matrix is None else np.asarray(matrix)
    eigenvalues, eigenvectors = np.linalg.eig(matrix_array)
    return eigenvalues, eigenvectors


def main():
    print("--- Vector Operations ---")
    vector_sum, vector_difference, dot_product = vector_operations()
    print(f"Sum of v1 and v2: {vector_sum}")
    print(f"Difference of v1 and v2: {vector_difference}")
    print(f"Dot product of v1 and v2: {dot_product}")
    print("-" * 25)

    print("\n--- Matrix Operations ---")
    matrix_sum, matrix_product = matrix_operations()
    print(f"Sum of M1 and M2:\n{matrix_sum}")
    print(f"Product of M1 and M2:\n{matrix_product}")
    print("-" * 25)

    print("\n--- Eigen-analysis ---")
    eigenvalues, eigenvectors = eigen_analysis()
    print(f"Matrix A:\n{np.array([[4, 1], [2, 3]])}")
    print(f"Eigenvalues: {eigenvalues}")
    print(f"Eigenvectors:\n{eigenvectors}")
    print("-" * 25)


if __name__ == "__main__":
    main()