Scalars, Vectors, Matrices

A scalar is a single number. A vector is a 1D array (features of one sample). A matrix is 2D (rows = samples, cols = features). Tensors are higher-dim arrays (images, sequences).

import numpy as np

scalar = 5
vector = np.array([2, 5, 8])
matrix = np.array([
    [22, 1, 0],
    [38, 1, 1],
    [26, 0, 0],
    [35, 1, 1]
])

print("Scalar:", scalar)
print("Vector:", vector)
print("Matrix:\\n", matrix)

Matrix Multiplication & Dot Product

The dot product measures alignment/similarity of two vectors. In ML, predictions often compute y = w · x (weights dot features). Matrix multiplication stacks many dot products.

# Linear model prediction for one sample
x = np.array([22, 1, 0])      # features: e.g., [Age, Sex_male, Survived?]
w = np.array([0.03, 0.7, 0.0])# weights learned by the model
y_hat = np.dot(w, x)          # model score
print("Prediction score:", y_hat)

# Batch prediction: XW for many samples
X = matrix[:, :2]             # take first 2 columns as features
W = np.array([[0.03], [0.7]]) # weights as a column vector
scores = X @ W                # matrix multiply (same as np.matmul)
print("Scores:\\n", scores)

Cosine Similarity (Vector Similarity)

Used in recommendation/NLP. Values near 1 mean “very similar direction”.

u = np.array([1, 2, 3])
v = np.array([2, 4, 6])
cos_sim = np.dot(u, v) / (np.linalg.norm(u) * np.linalg.norm(v))
print("Cosine similarity:", cos_sim)

Eigenvalues & Eigenvectors (PCA intuition)

Eigenvectors give principal directions of variance; eigenvalues tell how much variance lies along them. PCA projects data onto top eigenvectors to reduce dimensions with minimal information loss.

# Covariance matrix and eigen-decomposition
X = np.array([[22, 7.25],
              [38, 71.28],
              [26, 7.93],
              [35, 53.10]], dtype=float)  # [Age, Fare]
X = (X - X.mean(axis=0)) / X.std(axis=0)  # standardize
cov = np.cov(X.T)
vals, vecs = np.linalg.eig(cov)
print("Eigenvalues:", vals)
print("Eigenvectors:\\n", vecs)

Distributions (quick tour)

• Normal: continuous, bell-curve (heights, measurement noise)
• Bernoulli: single yes/no trial (clicked ad or not)
• Binomial: # of successes in n Bernoulli trials (k clicks in n shows)
• Poisson: event counts in fixed time/space (support tickets/hour)

import numpy as np
from scipy.stats import norm, bernoulli, binom, poisson

# Normal(0,1): mean ≈ 0, std ≈ 1 as n grows
samples = np.random.normal(loc=0, scale=1, size=10000)
print("Mean≈", round(samples.mean(), 3), "Std≈", round(samples.std(), 3))

# Bernoulli trials (p=0.3)
print("Bernoulli:", bernoulli.rvs(p=0.3, size=10))

# Binomial: P(X = 5) when n=10, p=0.5
from math import comb
n, p, k = 10, 0.5, 5
pmf_5 = comb(n,k) * (p**k) * ((1-p)**(n-k))
print("Binomial P(X=5):", pmf_5)

# Poisson samples with lambda=3
print("Poisson:", poisson.rvs(mu=3, size=5))

Conditional Probability & Bayes' Theorem

P(A|B) = P(B|A)·P(A) / P(B). In spam filtering, A=spam, B=“contains ‘win’”.

# Simple Bayes example (toy numbers)
P_spam = 0.2
P_word_given_spam = 0.9
P_word = 0.4
P_spam_given_word = (P_word_given_spam * P_spam) / P_word
print("P(spam | word):", round(P_spam_given_word, 3))

Descriptive Stats

import numpy as np
import pandas as pd
from scipy import stats

data = [7, 8, 5, 6, 9, 10, 6, 7, 8]

print("Mean:", np.mean(data))
print("Median:", np.median(data))
print("Mode:", stats.mode(data, keepdims=True)[0][0])
print("Variance:", np.var(data, ddof=0))     # population variance
print("Std Dev:", np.std(data, ddof=0))      # population std

Correlation & Covariance

df = pd.DataFrame({
    "Study_Hours": [2, 3, 4, 5, 6],
    "Exam_Score":  [50, 60, 65, 70, 80]
})
print("Covariance:\\n", df.cov())
print("Correlation:\\n", df.corr())

Hypothesis Testing (one-sample t-test)

Test if the mean differs from 7 (e.g., average satisfaction score).

t_stat, p_value = stats.ttest_1samp(data, popmean=7)
print("t-stat:", round(t_stat,3), "p-value:", round(p_value,4))
# If p < 0.05, we reject H0 (mean == 7) at 5% significance.