4.1.1 Linear Algebra Roadmap: Data as Vectors, Batches as Matrices

Linear algebra is the language AI uses to represent data and transform it. Do not start by memorizing proofs. First see what each object does in code.

Look at the Map First

The chapter flow is:

Idea	First meaning in AI
vector	one object written as numbers
matrix	many vectors stacked together, or a transformation
dot product	compare matching positions and add them up
matrix multiplication	many dot products at once
eigenvalue / eigenvector	important directions, useful for PCA intuition

Run the Smallest Loop

Create linear_algebra_first_loop.py and run it after installing numpy.

import numpy as np

student = np.array([90, 85, 92])
students = np.array(
    [
        [90, 85, 92],
        [70, 88, 75],
        [95, 91, 89],
    ]
)
weights = np.array([0.4, 0.2, 0.4])

single_score = student @ weights
all_scores = students @ weights

print("student_vector:", student)
print("matrix_shape:", students.shape)
print("single_score:", round(single_score, 2))
print("all_scores:", all_scores.round(2))

Expected output:

student_vector: [90 85 92]
matrix_shape: (3, 3)
single_score: 89.8
all_scores: [89.8 75.6 91.8]

If you accidentally use * instead of @, you get element-by-element multiplication, not a weighted score. This is the most useful early distinction.

Learn in This Order

Order	Read	What to focus on first
1	4.1.2 Vectors	object -> vector, length, dot product, cosine similarity
2	4.1.3 Matrices	batch data, matrix multiplication, X @ W + b
3	4.1.4 Eigenvalues and Eigenvectors	special directions, PCA intuition
4	4.1.5 Vector Spaces	basis, dimension, linear transformation

Pass Check

You pass this roadmap when you can explain why one sample is a vector, why a batch is a matrix, what @ does, and why these ideas reappear in RAG similarity, PCA, and neural network layers.