Linear Algebra for Data Science: Vectors & Matrices

Interpret matrix and vector methods before reaching for a formula

In linear regression, predictions can be written compactly as Xw: the feature matrix X multiplied by the parameter vector w.

• Vectors represent ordered quantities, matrices organize transformations and datasets, and matrix multiplication composes linear operations.
• Dot products measure alignment and appear in projections, similarity and neural-network layers.
• Eigenvectors identify directions preserved by a linear transformation, while eigenvalues describe the scaling along those directions.
• Shape checking is essential: many linear-algebra bugs are dimension mismatches disguised by broadcasting.

Calculate a small case and verify it in Python

For matrix and vector methods, predict one visible result before each snippet runs—a row count, shape, statistic, ranking or error pattern—then compare it with the printed checkpoint.

# matrix and vector methods · example 1
import numpy as np
X = np.array([[1, 2], [1, 5], [1, 8]], dtype=float)
w = np.array([10, 3], dtype=float)
y_hat = X @ w
print(y_hat)

matrix and vector methods checkpoint 1: Run the cell and inspect the printed values before changing anything.

# matrix and vector methods · example 2
import pandas as pd
import numpy as np

df = pd.read_csv('practice.csv')
x = df.select_dtypes('number').iloc[:,0].dropna()
print('n=', len(x), 'mean=', round(x.mean(),2), 'median=', round(x.median(),2))

matrix and vector methods checkpoint 2: n, mean and median are printed from the first numeric column.

# matrix and vector methods · example 3
q1, q3 = x.quantile([0.25,0.75])
iqr = q3 - q1
print('std=', round(x.std(ddof=1),2), 'IQR=', round(iqr,2))

matrix and vector methods checkpoint 3: Standard deviation and IQR provide two different views of spread.

# matrix and vector methods · example 4
rng = np.random.default_rng(42)
boot = [rng.choice(x, size=len(x), replace=True).mean() for _ in range(1000)]
print(np.percentile(boot,[2.5,50,97.5]).round(2))

matrix and vector methods checkpoint 4: The three values are the bootstrap 2.5th, 50th and 97.5th percentiles.

# matrix and vector methods · example 5
x2 = pd.concat([x, pd.Series([x.max()*5])], ignore_index=True)
print('mean change:', round(x2.mean()-x.mean(),2))
print('median change:', round(x2.median()-x.median(),2))

matrix and vector methods checkpoint 5: The mean usually moves more than the median when one extreme value is added.

Stress-test the result against assumptions and outliers

Self-check with answers

Practise interpretation, not only arithmetic