Cheatsheet: Mathematics for Machine Learning 1: Linear Algebra Foundations

By Better Wise

One-page lookup for vectors, matrices, projections, eigen and SVD, with every operation in NumPy.

import numpy as np

Module 1: Vectors, Norms and Similarity

1.1 Vectors as Arrows, Points, and NumPy Arrays

A vector is the same object seen three ways: an arrow (direction + length), a point (coordinates), and a 1D array (data).

View	Meaning	NumPy
Arrow	Direction and magnitude from origin	`v = np.array([3, 4])`
Point	A location in space	`v[0], v[1]` are coordinates
Array	Ordered list of numbers	`v.shape # (2,)`

v = np.array([3, 4])
u = np.array([1, 2])
v + u            # elementwise add (tip-to-tail)
v - u            # difference vector
2 * v            # scalar scaling (stretch)
np.linalg.norm(v)  # length

Tip-to-tail: v + u places u at the tip of v.
Scalar multiple c * v: same direction if c > 0, flipped if c < 0, zero if c = 0.
Dimension = v.shape[0]. Two vectors add only if dimensions match.

1.2 Norms and Distances

Norm = "size" of a vector. Distance = norm of the difference.

Norm	Formula	NumPy	Shape it measures
L2 (Euclidean)	$\sqrt{\sum v_i^2}$	`np.linalg.norm(v)`	Straight-line length
L1 (Manhattan)	$\sum \lvert v_i \rvert$	`np.linalg.norm(v, 1)`	Grid / city-block
L-infinity	$\max \lvert v_i \rvert$	`np.linalg.norm(v, np.inf)`	Largest component

$$ |v|_2 = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2} $$

Distance	Formula	NumPy
Euclidean	$\|u - v\|_2$	`np.linalg.norm(u - v)`
Manhattan	$\|u - v\|_1$	`np.linalg.norm(u - v, 1)`

Unit vector (length 1): v / np.linalg.norm(v). Normalizing puts every vector on the unit circle/sphere.

1.3 Dot Product and Cosine Similarity

$$ u \cdot v = \sum_i u_i v_i = |u|,|v|\cos\theta $$

np.dot(u, v)      # or u @ v

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