5.3 Dimension

Introduction

One dimension is a line, two dimensions is a plane, three dimensions is the space we live in – everyone has an intuitive sense of this. But how do we define “dimension” rigorously in mathematics?

Dimension is the minimum number of independent directions needed to describe a space. In linear algebra, this number is precisely the number of vectors in a basis.

Here is the remarkable fact: no matter which basis you choose for a given vector space, the number of basis vectors is always the same. This is the theorem that makes dimension a well-defined concept.

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Definition of Dimension

The dimension of a vector space $V$ is the number of vectors in any basis for $V$:

\[\dim(V) = \text{(number of vectors in a basis)}\]

This number does not depend on which basis you choose.

Special case: The zero space ${\vec{0}}$ has the empty set as its basis, so

\[\dim(\{\vec{0}\}) = 0\]

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Familiar Examples

$\mathbb{R}^1$ – The Number Line

Basis: ${1}$ (or any single nonzero real number) \(\dim(\mathbb{R}^1) = 1\)

$\mathbb{R}^2$ – The Plane

Basis: ${\hat{e}_1, \hat{e}_2} = {(1,0)^\top, (0,1)^\top}$ \(\dim(\mathbb{R}^2) = 2\)

Another valid basis: ${(1,1)^\top, (1,-1)^\top}$ – these vectors are linearly independent and span $\mathbb{R}^2$, so they form a legitimate basis.

$\mathbb{R}^3$ – Three-Dimensional Space

Basis: ${\hat{e}_1, \hat{e}_2, \hat{e}_3}$ \(\dim(\mathbb{R}^3) = 3\)

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Dimension Table

Space	Example Basis	Dimension
${\vec{0}}$	Empty set $\emptyset$	$0$
Line through the origin	One vector along the line	$1$
Plane through the origin	Two non-parallel vectors	$2$
$\mathbb{R}^2$	${\hat{e}_1, \hat{e}_2}$	$2$
$\mathbb{R}^3$	${\hat{e}_1, \hat{e}_2, \hat{e}_3}$	$3$
$\mathbb{R}^n$	${\hat{e}_1, \ldots, \hat{e}_n}$	$n$

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The Dimension Theorem

Theorem: Every basis for a finite-dimensional vector space $V$ contains the same number of vectors.

Without this theorem, the very definition of dimension would be ambiguous. The key idea behind the proof: assuming two bases of different sizes leads to a contradiction.

Corollaries:

• Any set of $\dim(V)$ linearly independent vectors in $V$ is automatically a basis.
• Any set of $\dim(V)$ vectors that spans $V$ is automatically a basis.

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Dimension of Subspaces

If $W$ is a subspace of $V$, then

\[\dim(W) \leq \dim(V)\]

Equality holds if and only if $W = V$.

Example: Subspaces of $\mathbb{R}^3$:

Subspace	Geometric Meaning	Dimension
${\vec{0}}$	The origin	$0$
$\text{span}{\vec{v}}$	Line through the origin	$1$
$\text{span}{\vec{v}_1, \vec{v}_2}$	Plane through the origin	$2$
$\mathbb{R}^3$ itself	Full 3D space	$3$

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Extending and Reducing to a Basis

Basis extension theorem: A basis $\mathcal{B}_W$ for a subspace $W$ can always be extended to a basis $\mathcal{B}_V$ for the full space $V$.

Extending a linearly independent set: You can keep adding vectors to a linearly independent set (while preserving independence) until you reach a basis.

Reducing a spanning set: You can keep removing redundant vectors from a spanning set (while preserving the span) until you arrive at a basis.

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Intuitive Understanding

Think of dimension as the number of degrees of freedom.

• To specify a point on a line (1D space), you need one number.
• To specify a point on a plane (2D space), you need two numbers.
• To specify a point in $n$-dimensional space, you need $n$ numbers.

Number of basis vectors = number of independent numbers needed to describe the space = dimension.

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Key Takeaways

Concept	Description / Formula
Definition of dimension	$\dim(V)$ = number of vectors in a basis
Dimension of the zero space	$\dim({\vec{0}}) = 0$
Dimension of $\mathbb{R}^n$	$\dim(\mathbb{R}^n) = n$
Dimension theorem	Every basis has the same number of vectors
Subspace dimension	$\dim(W) \leq \dim(V)$
Intuition	Dimension = number of numbers needed to specify a point in the space

Next up: Null Space and Column Space – when does the matrix equation $A\vec{x}=\vec{b}$ have a solution?