Vectors and Vector Spaces: An Axiomatic Approach

In modern mathematics, a vector is not merely a directed line segment but an element of an algebraic structure called a Vector Space. This lesson develops the theory of finite and infinite-dimensional vector spaces, focusing on the structural properties that define linear algebra.

1. Formal Axioms of Vector Spaces

Let $F$ be a field (typically $\mathbb{R}$ or $\mathbb{C}$). A Vector Space over $F$ is a set $V$ equipped with two operations: binary addition $+:V\times V\to V$ and scalar multiplication $\cdot :F\times V\to V$, satisfying the following eight axioms for all $u,v,w\in V$ and $a,b\in F$:

1. Additive Commutativity: $u+v=v+u$.
2. Additive Associativity: $(u+v)+w=u+(v+w)$.
3. Additive Identity: There exists an element $0\in V$ such that $v+0=v$.
4. Additive Inverse: For every $v\in V$, there exists $-v\in V$ such that $v+(-v)=0$.
5. Multiplicative Identity: $1\cdot v=v$, where $1$ is the multiplicative identity of $F$.
6. Compatibility of Scalar Multiplication: $a(bv)=(ab)v$.
7. Distributivity over Vector Addition: $a(u+v)=au+av$.
8. Distributivity over Scalar Addition: $(a+b)v=av+bv$.

These axioms imply that $V$ is an abelian group under addition and that scalar multiplication behaves linearly.

2. Linear Independence and the Steinitz Exchange Lemma

A set of vectors $\{v_1,\dots ,v_n\}$ is linearly independent if the only solution to ${\sum }_{i=1}^n{c}_i{v}_i=0$ is $c_i=0$ for all $i$. If a set is not linearly independent, it is linearly dependent.

The Steinitz Exchange Lemma

This fundamental lemma provides the backbone for the theory of dimension. Statement: Let $\{u_1,u_2,\dots ,u_m\}$ be a linearly independent set of vectors in a vector space $V$, and let $\{v_1,v_2,\dots ,v_n\}$ be a spanning set for $V$. Then:

1. $m\le n$.
2. There exists a subset of the spanning set of size $n-m$ which, when added to the independent set, still spans $V$.

Implication: Every basis of a finite-dimensional vector space must have the same number of elements.

3. Basis and Dimension

A Basis $\mathcal{B}$ for $V$ is a linearly independent set that spans $V$. The Dimension of $V$, denoted $\dim (V)$, is defined as the cardinality of its basis.

Invariance of Dimension: By the Steinitz Lemma, any two bases of $V$ have the same cardinality.

• If $\dim (V)=n$, then any $n$ linearly independent vectors form a basis.
• Any spanning set with $n$ vectors is a basis.
• A subspace $W\subseteq V$ satisfies $\dim (W)\le \dim (V)$, with equality if and only if $W=V$.

4. Subspaces, Quotients, and Isomorphism

A subset $W\subseteq V$ is a subspace if $0\in W$ and $W$ is closed under addition and scalar multiplication.

Quotient Spaces

Given a subspace $W\subseteq V$, the Quotient Space $V/W$ is the set of cosets $v+W=\{v+w:w\in W\}$. Addition and multiplication are defined as:

• $(u+W)+(v+W)=(u+v)+W$
• $c(v+W)=(cv)+W$

The dimension of the quotient space is given by the formula: $\dim (V/W)=\dim (V)-\dim (W)$

The First Isomorphism Theorem

Let $T:V\to U$ be a linear transformation. Then: $V/\ker (T)\cong \text{im}(T)$ This relates the structure of the domain, the kernel (null space), and the image (range).

5. Direct Sums and Projections

A vector space $V$ is the Direct Sum of two subspaces $U$ and $W$, denoted $V=U\oplus W$, if every $v\in V$ can be uniquely written as $v=u+w$ with $u\in U$ and $w\in W$. This occurs if and only if $V=U+W$ and $U\cap W=\{0\}$.

A Projection is a linear operator $P:V\to V$ such that $P^2=P$. Every projection defines a direct sum decomposition $V=\text{im}(P)\oplus \ker (P)$.

6. The Dual Space $V^{\ast }$

For a vector space $V$ over $F$, the Dual Space $V^{\ast }$ is the set of all linear functionals $f:V\to F$. If $V$ has a basis $\{e_1,\dots ,e_n\}$, the Dual Basis $\{{\varphi }_1,\dots ,{\varphi }_n\}$ is defined such that: ${\varphi }_i(e_j)={\delta }_{ij}$ where ${\delta }_{ij}$ is the Kronecker delta. Note that $\dim (V^{\ast })=\dim (V)$ in finite dimensions, but $V$ is only naturally isomorphic to its double-dual $V^{\ast \ast }$.

7. Change of Basis and Transition Operators

Consider two bases for $V$: $\mathcal{B}=\{b_1,\dots ,b_n\}$ and $\mathcal{C}=\{c_1,\dots ,c_n\}$. Any vector $v$ has coordinates $[v{]}_{\mathcal{B}}$ and $[v{]}_{\mathcal{C}}$. The Transition Matrix $P_{\mathcal{C}\leftarrow \mathcal{B}}$ (or $P$) is defined such that: $[v{]}_{\mathcal{C}}=P_{\mathcal{C}\leftarrow \mathcal{B}}[v{]}_{\mathcal{B}}$ The $j$-th column of $P$ is $[b_j{]}_{\mathcal{C}}$.

8. Infinite Dimensional Spaces

In infinite dimensions, the concept of a basis becomes more subtle:

• Hamel Basis: A set such that every vector is a finite linear combination of basis elements. Using the Axiom of Choice (Zorn’s Lemma), every vector space has a Hamel basis.
• Schauder Basis: In a normed vector space, a sequence $\{e_n\}$ such that every vector has a unique representation as a convergent infinite series $v={\sum }_{n=1}^{\infty }{a}_n{e}_n$. This requires a topology.

Python Implementation: Rank and Change of Basis

We use numpy for numerical rank computation and sympy for exact symbolic basis transformations.

import numpy as np
import sympy as sp

# 1. Linear Independence check using NumPy
def check_linear_independence(vectors):
    matrix = np.array(vectors)
    rank = np.linalg.matrix_rank(matrix)
    is_independent = rank == len(vectors)
    return rank, is_independent

vecs = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] # Dependent: 3rd = 2*2nd - 1st
rank, independent = check_linear_independence(vecs)
print(f"Rank: {rank}, Independent: {independent}")

# 2. Symbolic Change of Basis using SymPy
# Define Basis B (standard) and Basis C
B = sp.eye(3)
C = sp.Matrix([[1, 1, 0], [1, 0, 1], [0, 1, 1]])

# Transition Matrix from B to C is C^-1 * B
P_B_to_C = C.inv()

v_B = sp.Matrix([1, 2, 3])
v_C = P_B_to_C * v_B
print(f"Coordinates in Basis C: {v_C}")