The Axiomatic Foundation of Vector Spaces

In elementary physics, a vector is often described as a directed line segment. While intuitive, this definition is insufficient for higher mathematics. Modern linear algebra treats a Vector Space as an abstract algebraic structure—a “playground” where elements can be added together and scaled by numbers.

1. Defining the Playground

A Vector Space $V$ over a field $F$ (typically $\mathbb{R}$) is a set equipped with two operations: vector addition ($+$) and scalar multiplication ($\cdot$). Instead of memorizing axioms as dry rules, we can view them as the “laws of physics” for our data.

Standard Euclidean Space: $\mathbb{R}^n$

The most common example is $\mathbb{R}^n$, where addition and scaling are performed component-wise. Let’s verify the Commutativity ($u + v = v + u$) and Distributivity ($a(u+v) = au + av$) properties using NumPy.

import numpy as np

# Define two vectors in R3
u = np.array([1, 2, 3])
v = np.array([4, 5, 6])
a = 2.5

# 1. Verify Commutativity
lhs_add = u + v
rhs_add = v + u
print(f"Addition Commutativity: {lhs_add} == {rhs_add}")

# 2. Verify Distributivity
lhs_dist = a * (u + v)
rhs_dist = a * u + a * v
print(f"Distributivity: {lhs_dist} == {rhs_dist}")

# Check if they are numerically equal
print(f"Equal? {np.allclose(lhs_dist, rhs_dist)}")

2. The Eight Axioms

For any set $V$ to be a formal Vector Space, these eight rules must hold for all $u, v, w \in V$ and scalars $a, b \in \mathbb{R}$:

1. Commutativity: $u + v = v + u$.
2. Associativity: $(u + v) + w = u + (v + w)$.
3. Additive Identity: There is a $\mathbf{0}$ such that $v + \mathbf{0} = v$.
4. Additive Inverse: For every $v$, there is a $-v$ such that $v + (-v) = \mathbf{0}$.
5. Multiplicative Identity: $1 \cdot v = v$.
6. Compatibility: $a(bv) = (ab)v$.
7. Distributivity of Scalar: $a(u + v) = au + av$.
8. Distributivity of Vector: $(a + b)v = av + bv$.

Exercise: The Non-Vector Space

Consider the set of all points in the first quadrant: $Q = \{(x, y) \in \mathbb{R}^2 \mid x, y \ge 0\}$. Why does this fail to be a vector space?

import numpy as np

# A point in the first quadrant
v = np.array([5, 10])

# Scaling by a negative number
scalar = -1
result = scalar * v

print(f"Original vector: {v}")
print(f"Scaled by -1: {result}")

# Is result in the first quadrant?
is_in_q = np.all(result >= 0)
print(f"Is the result still in the first quadrant? {is_in_q}")

The failure shown above violates Closure under Scalar Multiplication. If we scale a “positive” vector by a negative number, we leave the set. Thus, the first quadrant is not a vector space.

3. Abstract Examples: Polynomials

The beauty of these axioms is that “vectors” don’t have to be arrows. They can be functions or polynomials. The set $\mathbb{P}_n$ of polynomials of degree $\le n$ forms a vector space because adding two polynomials yields another polynomial, and the axioms hold.

4. Function Spaces

In advanced applications like Fourier analysis, we treat signals (functions) as vectors. If $f(t)$ and $g(t)$ are continuous functions, then $h(t) = f(t) + g(t)$ is also continuous.

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, 1, 100)
f = np.sin(2 * np.pi * t)
g = np.cos(4 * np.pi * t)

# Vector addition of 'functions' (sampled)
h = f + g

plt.plot(t, f, '--', label='f(t)')
plt.plot(t, g, '--', label='g(t)')
plt.plot(t, h, 'k', linewidth=2, label='f(t) + g(t)')
plt.legend()
plt.title("Addition in Function Space")
plt.show()

5. Summary Check