In machine learning projects, I sometimes run into a situation where I have to write math formulas.

I created this for my reference but wanted to share with you all how to write basic math formulas in python.

Indexing

\[x_i\]

is the \(i^{th}\) value/element of a vector x.

x = [10, 20, 30]
i = 0
print(x[i]) # = 10

for i in range(len(x)):
  print(x[i])

This can be represented in a 2D vectors. \(x_{ij}\)

x = [ [10, 20, 30], [40, 50, 60] ]
i = 0
j = 0
print(x[i][j])  # = 10

for i in range(len(x)):
  for j in range(len(x[i])):
    print(x[i][j])

Sigma

\[\sum^N_{i=1}x_i\]

This formula represents the sum of all elements of a vector within a certain range from lower limit \(i=1\) to upper limit \(N\) In python, it same as iterating from index 0 to index N-1

x = [1, 2, 3, 4, 5]
result = 0
N = len(x)
for i in range(N):
  result += x[i]  # result = result + x[i]
print(result) # = 15

We can represent the code above by using built in functions as follows:

x = [1, 2, 3, 4, 5]
result = sum(x)

Average

\[\frac{1}{N}\sum^N_{i=1}x_i\]

Here, we calculate the average by dividing the Sigma we saw above by the number of elements in the vector.

x = [1, 2, 3, 4, 5]
result = 0
N = len(x)
for i in range(N):
  result = result + x[i]
average = result / N
print(average)

which can be simlified as:

x = [1, 2, 3, 4, 5]
result = sum(x) / len(x)

PI

\[\Pi^N_{i=1}x_i\]

The next formula represents the product of all elements of a vecotor within a range.

x = [1, 2, 3, 4, 5]
result = 1
N = len(x)
for i in range(N):
  result *= x[i]  # result = result * x[i]
print(result)

Absolute Value

\(\Vert x\Vert\)
\(\Vert y\Vert\)

The next formula indicates the absolute value.

x = 10
y = -20
abs(x)  # 10
abs(y)  # 20

Norm of vector

\[\Vert x\Vert\]

Norm is used to calculate the size of a vector. In python, it’s the sum of the square of every element and taking a square root of it.

import math

x = [1, 2, 3]
math.sqrt(x[0]**2 + x[1]**2 + x[2]**2)
# or
math.sqrt(sum([v**2 for v in x]))

Function

\[f: X \rightarrow Y\]

Function takes a domain of X and map it to Y. In python, it calls the values of X and computes the values of Y.

def f(X):
  Y = ...
  return Y

\[f:R \rightarrow R\]

R indicates that the input and the output are real numbers which can be any of the following: integer, float, irrational, rational. In python, it any values excluding the complex numbers.

import math
x = 1
y = 2.5
z = math.pi

def f(k):
  k = ...
  return k

\[f:R^d \rightarrow R\]

$R^d$ is a d-dimensional vector of a real number. If d = 2, we would take a 2-D list and return the sum of the elements by using a function in python. This represent mapping $R^d$ to $R$.

X = [1,2]
f = sum
Y = f(X)

Tensor

Transpose

\[X^T\]

This changes the rows and columns of a matrix. In python:

import numpy as np
X = [[1, 2, 3],
    [4, 5, 6]]
np.transpose(X)

[[1,4],
 [2,5],
 [3,6]]

Element wise multiplication

\[z = x \odot y\]

This represents the multiplication of elements of two tensors. In python, we multiply the elements of the two lists.

import numpy as np
x = [[1,2],
    [3,4]]
y = [[2,2],
    [2,2]]
z = np.multiply(x,y)

Output:

[[2 4]
 [6 8]]

Dot Product

\(xy\)
\(x \cdot y\)

This is an inner product (aka dot product). The two vectors should have the same dimension. \[x = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, y = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}\] \[x^Ty = [1 \; 2 \; 3] \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32\]

x = [1, 2, 3]
y = [4, 5, 6]
dot = sum([i*j for i, j in zip(x, y)])
# 1*4 + 2*5 + 3*6
# 32

Exclamation

\[x!\]

Factorial is a product of 1 through the number. In python:

x = 5
fact = 1
for i in range(x, 0, -1):
    fact *= i # fact = fact * i
print(fact)

There’s a built-in function for this as well.

import math
x = 5
math.factorial(x)

Output:

# 5*4*3*2*1
120