Vector Algebra ’from scratch” with type annotations

Vector Algebra ‘from scratch’ with type annotations

I looked into Joel Grus’ Data science from Scratch and found his approach really nice as he actively uses type annotations and assertions in his fuctions.

So, below I practice vector operation functions from the book.

from typing import List
Vector = List[float]
Vector

typing.List[float]

Vector addition and subtraction

$ + = \[\begin{bmatrix} v_1 + w_1 \\ \ldots \\ v_n + w_n \end{bmatrix}\]

$

def add(vector1: Vector, vector2: Vector) -> Vector:
    assert len(vector1) == len(vector2)
    return [v1 + v2 for v1, v2 in zip(vector1, vector2)]

assert add([1, 2, 3], [4, 5, 6]) == [5, 7, 9]

$ - = \[\begin{bmatrix} w_1 - v_1 \\ \ldots \\ w_n - v_n \end{bmatrix}\]

$

def subtract(vector1: Vector, vector2:Vector) -> Vector:
    assert len(vector1) == len(vector2)
    return [v1 - v2 for v1, v2 in zip(vector1, vector2)]

assert subtract([1, 2, 3], [4, 5, 6]) == [-3, -3, -3]

Vector sum

$ + + = \[\begin{bmatrix} v_1 + \ldots + w_1 \\ \ldots \\ v_n + \ldots + w_n \end{bmatrix}\]

$

def vector_sum(vectors: List[Vector]) -> Vector:
    assert vectors
    
    vector_length = len(vectors[0])
    assert all(len(v) == vector_length for v in vectors)

    sums = [0] * vector_length
    for vector in vectors:
        sums = add(sums, vector)

    return sums

assert vector_sum([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) == [12, 15, 18]    

Scalar multiplication

$ c = \[\begin{bmatrix} c \cdot v_0 \\ \ldots \\ c \cdot v_n \end{bmatrix}\]

$

def scalar_multiply(c: float, vector: Vector) -> Vector:
    return [c * v for v in vector]

assert scalar_multiply(4, [1, 2, 3]) == [4, 8, 12]

Vector mean

$ (, , ) = \[\begin{bmatrix} \frac {v_1 + \ldots + w_1}{n} \\ \ldots \\ \frac{v_n + \ldots + w_n} {n} \end{bmatrix}\]

$

def vector_mean(vectors: List[Vector]) -> Vector:
    n = len(vectors)
    return scalar_multiply(1/n, vector_sum(vectors))

assert vector_mean([[1, 2], [3, 4], [5, 6]]) == [3, 4]

Dot product

$ = _{i=1}^{n} v_1 w_1 + + v_n w_n $

def dot(vector1: Vector, vector2: Vector) -> float:
    assert len(vector1) == len(vector2)

    return sum(v1 * v2 for v1, v2 in zip(vector1, vector2))


assert dot([1, 2, 3], [4, 5, 6]) == 32

Vector sum of squares

$ = _{i=1}^n v_1^2 + + v_n^2 $

def sum_of_squares(v: Vector) -> Vector:
    return dot(v, v)

assert sum_of_squares([1, 2, 3]) == 14

Magnitude

$ || || = $

import math as m

def magnitude(v: Vector) -> Vector:
    return m.sqrt(sum_of_squares(v))

assert magnitude([1, 2, 3]) == m.sqrt(14)

Distance of two vectors

$ d = $

def distance(vector1: Vector, vector2: Vector) -> Vector:
    return magnitude(subtract(vector1, vector2))