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.

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?

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.

5. Summary Check