By Bernd Klein. Last modified: 03 Feb 2025.

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Definitions

A linear combination in mathematics is an expression constructed from a set of terms by multiplying each term by a constant and adding the results.

Example of a linear combination:

p = λ₁ · x₁ + λ₂ · x₂ … λ_n · x_n

p is the scalar product of the values x₁, x₂ … x_n and
λ₁, λ₂ … λ_n are called scalars.

In most applications x₁, x₂ … x_n are vectors and the lambdas are integers or real numbers. (For those, who prefer it more formally:

x₁, x₂ … x_n ∈ V and V is a vector space, and
λ₁, λ₂ … λ_n ∈ K with K being a field)

Linear Combinations in Python

The vector y = (3.21, 1.77, 3.65) can be easily written as a linear combination of the unit vectors (0,0,1), (0,1,0) and (1,0,0):

(3.21, 1.77, 3.65) = 3.21 · (1,0,0) + 1.77 (0,1,0) + 3.65 · (0,0,1)

We can do the calculation with Python, using the module numpy:

import numpy as np
x = np.array([[0, 0, 1],
              [0, 1, 0],
              [1, 0, 0]])
y = ([3.65, 1.55, 3.42])
scalars = np.linalg.solve(x, y)
scalars

array([3.42, 1.55, 3.65])

The previous example was very easy, because we could work out the result in our head. What about writing our vector y = (3.21, 1.77, 3.65) as a linear combination of the vectors (0,1,1), (1,1,0) and (1,0,1)? It looks like this in Python:

import numpy as np
x = np.array([[0, 1, 1],
              [1, 1, 0],
              [1, 0, 1]])
y = ([3.65, 1.55, 3.42])
scalars = np.linalg.solve(x, y)
scalars

array([0.66, 0.89, 2.76])

Another Example

Any integer between -40 and 40 can be written as a linear combination of 1, 3, 9, 27 with scalars being elements of the set {-1, 0, 1}.

For example:

7 = 1 · 1 + (-1) · 3 + 1 · 9 + 0 · 27

We can calculate these scalars with Python. First we need a generator generating all the possible scalar combinations. If you have problems in understanding the concept of a generator, we recommend the chapter "Iterators and Generators" of our tutorial.

def factors_set():
    for i in [-1, 0, 1]:
        for j in [-1,0,1]:
            for k in [-1,0,1]:
                for l in [-1,0,1]:
                    yield (i, j, k, l)

We will use the memoize() technique (see chapter "Memoization and Decorators" of our tutorial) to memorize previous results:

def memoize(f):
    results = {}
    def helper(n):
        if n not in results:
            results[n] = f(n)
        return results[n]
    return helper

Finally, in our function linear_combination() we check every scalar tuple, if it can create the value n:

@memoize
def linear_combination(n):
    """ returns the tuple (i,j,k,l) satisfying
        n = i*1 + j*3 + k*9 + l*27      """
    weighs = (1,3,9,27)
    for factors in factors_set():
       sum = 0
       for i in range(len(factors)):
          sum += factors[i] * weighs[i]
       if sum == n:
          return factors

Putting it all together results in the following script:

def factors_set():
    for i in [-1, 0, 1]:
        for j in [-1, 0, 1]:
            for k in [-1, 0, 1]:
                for l in [-1, 0, 1]:
                    yield (i, j, k, l)
def memoize(f):
    results = {}
    def helper(n):
        if n not in results:
            results[n] = f(n)
        return results[n]
    return helper
@memoize
def linear_combination(n):
    """ returns the tuple (i,j,k,l) satisfying
        n = i*1 + j*3 + k*9 + l*27      """
    weighs = (1, 3, 9, 27)
    for factors in factors_set():
       sum = 0
       for i in range(len(factors)):
          sum += factors[i] * weighs[i]
       if sum == n:
          return factors
# calculate the linear combinations of the first 10 positive integers:
for i in range(1,11):
    print(linear_combination(i))

(1, 0, 0, 0)
(-1, 1, 0, 0)
(0, 1, 0, 0)
(1, 1, 0, 0)
(-1, -1, 1, 0)
(0, -1, 1, 0)
(1, -1, 1, 0)
(-1, 0, 1, 0)
(0, 0, 1, 0)
(1, 0, 1, 0)