Vector Spaces

Definition

A vector space $V$ over a field $K$ (usually $\mathbb{R}$ or $\mathbb{C}$) is a set with two operations:

• Addition: $\mathbf{u} + \mathbf{v} \in V$
• Scalar multiplication: $a\mathbf{u} \in V$ for $a \in K$

satisfying:

Property	Condition
Closure	$\mathbf{u} + \mathbf{v} \in V$, $a\mathbf{u} \in V$
Commutativity	$\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
Associativity	$(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
Zero vector	$\exists\,\mathbf{0}: \mathbf{u} + \mathbf{0} = \mathbf{u}$
Additive inverse	$\exists\,(-\mathbf{u}): \mathbf{u} + (-\mathbf{u}) = \mathbf{0}$
Distributivity 1	$a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}$
Distributivity 2	$(a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u}$
Compatibility	$a(b\mathbf{u}) = (ab)\mathbf{u}$
Identity	$1\mathbf{u} = \mathbf{u}$

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Examples

Vector Space	Elements	Field
$\mathbb{R}^n$	$n$-tuples $(x_1, \ldots, x_n)$	$\mathbb{R}$
Matrices $M_{m,n}$	$m \times n$ matrices	$\mathbb{R}$
Polynomials $P_n$	Degree $\leq n$ polynomials	$\mathbb{R}$
Functions $C[a,b]$	Continuous functions on $[a,b]$	$\mathbb{R}$

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Linear Combinations

A linear combination of vectors $\mathbf{v}_1, \ldots, \mathbf{v}_m$ is:

$a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \cdots + a_m\mathbf{v}_m$

where $a_i \in K$ are scalars.

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Linear Independence

Vectors $\mathbf{v}_1, \ldots, \mathbf{v}_m$ are linearly independent if:

$a_1\mathbf{v}_1 + \cdots + a_m\mathbf{v}_m = \mathbf{0} \Rightarrow a_1 = \cdots = a_m = 0$

Equivalently: no vector can be written as a combination of the others.

Test: Set up the linear combination equal to zero, solve for coefficients.

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Span

The span of vectors $\mathbf{v}_1, \ldots, \mathbf{v}_m$ is:

$\text{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_m\} = \{a_1\mathbf{v}_1 + \cdots + a_m\mathbf{v}_m : a_i \in K\}$

All linear combinations of the vectors.

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Basis and Dimension

A basis for $V$ is a set of vectors that:

1. Is linearly independent
2. Spans $V$

The dimension $\dim(V)$ is the number of vectors in any basis.

Key theorem: All bases have the same number of vectors.

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Example: $\mathbb{R}^3$

Standard basis: $\mathbf{e}_1 = (1,0,0)$, $\mathbf{e}_2 = (0,1,0)$, $\mathbf{e}_3 = (0,0,1)$

Any vector: $\mathbf{v} = v_1\mathbf{e}_1 + v_2\mathbf{e}_2 + v_3\mathbf{e}_3$

Dimension: $\dim(\mathbb{R}^3) = 3$

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Important Result

If $\mathbf{v}_1, \ldots, \mathbf{v}_m$ are linearly independent in $V$ and $\dim(V) = m$, then they form a basis.

In $\mathbb{R}^n$:

• More than $n$ vectors → linearly dependent
• Fewer than $n$ vectors → cannot span

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Why Vector Spaces Matter

• Unify disparate objects (vectors, matrices, functions) under one framework
• Enable general results applicable to all linear problems
• Foundation for quantum mechanics (Hilbert spaces)
• Underpin linear algebra (matrices represent linear maps between vector spaces)

The abstraction reveals common structure across seemingly different mathematical objects.