Vector Spaces

1. Introduction

A vector space is the central abstraction of linear algebra — a setting in which we can add objects together and scale them by numbers, while obeying a short list of natural rules. The objects need not be the familiar arrows of $\mathbb{R}^n$: they can be polynomials, matrices, continuous functions, or solutions of a differential equation. By stripping away everything except addition and scalar multiplication, the vector-space framework lets a single theory describe all of these at once. A theorem proved about abstract vector spaces applies instantly to every concrete example.

The genius of the theory is that a vast space can be reconstructed from a tiny, finite skeleton. A basis is a minimal set of vectors from which every other vector is built uniquely as a linear combination, and the size of that basis — the dimension — is an intrinsic number that pins down the space up to isomorphism. Concepts like linear independence, span, basis, dimension, and coordinates turn vague geometric intuition ("how many independent directions are there?") into precise, computable algebra.

This article develops the structure rigorously: the vector-space axioms, subspaces and the subspace test, linear combinations and span, linear independence, bases and the uniqueness of coordinates, dimension and its invariance, the four fundamental subspaces of a matrix, and the Rank–Nullity Theorem with proof. The aim is to move fluently between abstract definitions and concrete computations with row reduction.

The vector-space abstraction is what makes linear algebra universal: the same theorems govern arrows in $\mathbb{R}^3$, polynomials, matrices, and solutions of differential equations. Dimension is the precise answer to "how many independent degrees of freedom does this system have?" — a question that arises in physics, computer science, statistics, and engineering.

2. Core Concepts

2.1 The Vector Space Axioms

A vector space $V$ over a field $\mathbb{F}$ (usually $\mathbb{R}$) is a set with addition $\mathbf{u}+\mathbf{v}$ and scalar multiplication $c\mathbf{v}$ satisfying ten axioms: closure, commutativity, associativity, zero vector, additive inverses, distributivity, scalar associativity, and $1\mathbf{v}=\mathbf{v}$.

Familiar examples: $\mathbb{R}^n$; $P_n$ (polynomials of degree $\le n$); $M_{m\times n}$ (matrices); $C[a,b]$ (continuous functions).

2.2 Subspaces and the Subspace Test

A subspace $W\subseteq V$ is a subset that is itself a vector space. Verify with three conditions:

1. $\mathbf{0}\in W$;
2. Closed under addition;
3. Closed under scalar multiplication.

Examples: lines/planes through the origin; solutions of $A\mathbf{x}=\mathbf{0}$; symmetric matrices in $M_{n\times n}$.

2.3 Linear Combinations and Span

A linear combination is $c_1\mathbf{v}_1+\cdots+c_k\mathbf{v}_k$. The span is all such combinations: $\operatorname{span}\{\mathbf{v}_1,\dots,\mathbf{v}_k\} = \{\,c_1\mathbf{v}_1+\cdots+c_k\mathbf{v}_k : c_i\in\mathbb{F}\,\}.$ The span is always a subspace — the smallest subspace containing the given vectors.

2.4 Linear Independence

$\{\mathbf{v}_1,\dots,\mathbf{v}_k\}$ is linearly independent if $c_1\mathbf{v}_1+\cdots+c_k\mathbf{v}_k=\mathbf{0}$ implies all $c_i=0$. Otherwise the set is dependent, and at least one vector is redundant.

2.5 Basis and Coordinates

A basis is a set that is both independent and spanning. Every $\mathbf{v}\in V$ has a unique representation $\mathbf{v}=c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n$, and $(c_1,\dots,c_n)$ are the coordinates in that basis. Standard basis of $\mathbb{R}^3$: $\{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\}$. Standard basis of $P_2$: $\{1,x,x^2\}$.

2.6 Dimension

Every basis of a given $V$ has the same number of vectors; this is $\dim V$. Thus $\dim\mathbb{R}^n=n$, $\dim P_n=n+1$, $\dim M_{m\times n}=mn$. Consequences:

• Any independent set of $\dim V$ vectors is a basis.
• Any spanning set of $\dim V$ vectors is a basis.
• More than $\dim V$ vectors are always dependent; fewer cannot span.

Proof sketch of basis size invariance. If $\{\mathbf{v}_1,\dots,\mathbf{v}_m\}$ is independent and $\{\mathbf{w}_1,\dots,\mathbf{w}_n\}$ spans, then $m\le n$ (each $\mathbf{v}_i$ is a combination of the $\mathbf{w}_j$; independence forces $m\le n$). Reversing roles gives $n\le m$, so $m=n$.

2.7 The Four Fundamental Subspaces

For an $m\times n$ matrix $A$:

• Column space $\operatorname{Col}(A)\subseteq\mathbb{R}^m$: span of columns (range). Dimension = rank.
• Null space $\operatorname{Nul}(A)=\ker(A)\subseteq\mathbb{R}^n$: solutions of $A\mathbf{x}=\mathbf{0}$. Dimension = nullity.
• Row space $\subseteq\mathbb{R}^n$: span of rows; dimension = rank.
• Left null space: $\ker(A^{T})\subseteq\mathbb{R}^m$.

2.8 The Rank–Nullity Theorem

$\operatorname{rank}(A) + \dim\ker(A) = n \quad(\text{number of columns}).$

Proof. Row-reduce $A$ to RREF. The number of pivots is the rank $r$. Each pivot column has no free variable; each non-pivot column has one free variable. The null space is parametrized by the $n-r$ free variables, so $\dim\ker(A)=n-r$. Thus $r+(n-r)=n$.

Intuitively, each input dimension either survives into the output (rank) or collapses to zero (nullity).

2.9 Change of Basis

If $\mathcal{B}=\{\mathbf{v}_1,\dots,\mathbf{v}_n\}$ and $\mathcal{C}=\{\mathbf{w}_1,\dots,\mathbf{w}_n\}$ are two bases, the change-of-basis matrix $P_{\mathcal{C}\leftarrow\mathcal{B}}$ has columns giving coordinates of $\mathbf{v}_i$ in basis $\mathcal{C}$. Then $[\mathbf{x}]_{\mathcal{C}} = P_{\mathcal{C}\leftarrow\mathcal{B}}[\mathbf{x}]_{\mathcal{B}}$.