Finite-dimensional Vector Spaces

A vector space is one of the central objects of study in linear algebra. Any set which is compatible with the two operations of a vector space, namely vector addition and scaling, can be considered a vector space, and any element from a vector space is a vector.¹

To be compatible with vector scaling, a vector space must be accompanied with a field (or division ring for generality ²), typically denoted $𝖥$ or $𝖪$. This field may sometimes be called the base, ground, or underlying field, and an element from a field may be called a scalar. When the context is clear, the mention of a field may be omitted.

The smallest vector space contains only $\vec{0}$ and is called the trivial or zero vector space.

Abstract definition

A vector space over a field $𝖥$ is a commutative group $(\mathcal{V}, +)$ defined with a vector scaling function $𝖥 \times \mathcal{V} → \mathcal{V}$ of the form $(\alpha, \vec{x}) ↦ \alpha \vec{x}$, and with the following properties:

1. $1 \vec{x} = \vec{x}$
2. $\alpha (\beta \vec{x}) = (\alpha \beta) \vec{x}$
3. $(\alpha + \beta) \vec{x} = \alpha \vec{x} + \beta \vec{x}$
4. $\alpha(\vec{x} + \vec{y}) = \alpha \vec{x} + \alpha \vec{y}$

where $\alpha, \beta ∈ 𝖥$ and $\vec{x}, \vec{y} ∈ \mathcal{V}$.

Finite definition

Let $𝖥$ be a field, and let $𝖥^n$ be the set of all possible $n$-length lists of elements from $𝖥$. Then a finite $n$-dimensional vector space is one where $𝖥^n$ is defined with vector addition and scaling.

Finite Vector Addition

Let $[x_1 ⋯ x_n], [y_1 ⋯ y_n] ∈ 𝖥^n$. Finite vector addition can be defined:

The identity element of vector addition is called the zero vector or $\vec{0}$.

Finite Vector Scaling

Where $\alpha ∈ 𝖥$ and $[x_1 ⋯ x_n] ∈ 𝖥^n$, vector scaling can be defined:

Commentary

• A scalar is an element from a field $𝖥$.
• A vector is an element from a vector space.
• $𝖥^n$ is the set of all scalar lists of $n ∈ ℕ$ length.
• All finite $n$-dimensional vector spaces are linearly equivalent or isomorphic to $𝖥^n$.
• Any vector space which cannot be defined over finite lists is an infinite-dimensional vector space $(𝖥^∞)$.

It is conventional to assume that $𝖥^n$ is a shorthand reference for a finite $n$ dimensional vector space, and that $𝖥$ refers to the underlying field.

Linear Combination

Let $\mathcal{V} = \{ \vec{v}_1 ⋯ \vec{v}_{n} \}$ be a subset of a vector space, and let $\sigma_1 ⋯ \sigma_n$ be scalars from the underlying field. Then a linear combination of $\mathcal{V}$ is defined as any vector which is the sum of scaled vectors from $\mathcal{V}$:

A trivial combination is a linear combination where every scalar $\sigma_i$ is $0$.

A vanishing linear sum is any linear combination which results in $\vec{0}$.

Span

Let $\mathcal{V} = \{ \vec{v}_1 ⋯ \vec{v}_n \}$ be a subset of a vector space, and let $\sigma_1 ⋯ \sigma_n ∈ 𝖥$. Then the span of $\mathcal{V}$ is defined as the set of all linear combinations from $\mathcal{V}$.

If $\mathcal{K}$ is a subset of $\operatorname{span} (\mathcal{V})$ then we say $\mathcal{V}$ spans or generates $\mathcal{K}$.

The empty vector sum is defined as the additive identity $\vec{0}$, and thus the span of the empty set is the trivial vector space. Alternatively we can say that the span of any set generates the smallest vector space containing that set.

Linear Independence

Let $\mathcal{V} = \{ \vec{v}_1 ⋯ \vec{v}_n \}$ be a subset of a vector space, and let $\sigma_1 ⋯ \sigma_n ∈ 𝖥$. Then $\mathcal{V}$ is independent iff the trivial linear combination is the only sum which results in $\vec{0}$.

Conversely, $\mathcal{V}$ is dependent if any non-trivial combination of $\mathcal{V}$ can be $\vec{0}$.

Discussion

• $\mathcal{V}$ is independent iff the removal of any vector changes the span.
• $\mathcal{V}$ is independent iff no vector in $\mathcal{V}$ can be expressed as a combination of other vectors in $\mathcal{V}$.

Basis and Dimension

The basis of a vector space $\mathcal{V}$ is any independent set whose span is exactly $\mathcal{V}$. The dimension of $\mathcal{V}$ is the cardinality of any choice of basis, and is denoted $\dim(\mathcal{V})$.

Discussion

• All choices of basis for $\mathcal{V}$ have the same dimensionality.
• A basis may also be known as the minimal spanning or generating set.
• Any set which spans $\mathcal{V}$ but isn’t minimal can be trimmed down to a basis.
• Any independent subset of $\mathcal{V}$ which isn’t a basis can be expanded into a basis.
• For a finite vector space $𝖥^n$, any choice of basis for $𝖥^n$ will have the cardinality of $n$.
• A vector space is finite-dimensional if it has a finite basis.
• A finite-dimensional vector space has finite elements iff the underlying field is finite.

Linear Subspace

A linear subspace of the vector space $\mathcal{V}$ is any subset which is also a vector space under the same vector space operations of $\mathcal{V}$.

Discussion

• A subset of $\mathcal{V}$ is a linear subspace iff we have closure under addition and scalar multiplication.

Construction of Vector Spaces

Sum of Subspaces

Let $\mathcal{V}_1 ⋯ \mathcal{V}_n$ be subspaces of a vector space. Then the sum of subspaces is defined as a set which is closed under addition with a vector from each subspace.

$\mathcal{V}_1 ⋯ \mathcal{V}_n$ are independent subspaces if the intersection of any pair of subspaces is the trivial space. A direct sum is defined as the sum of independent subspaces, and is denoted:

Alternatively, $∑ \mathcal{V}_i$ is a direct sum iff only a trivial combination of vectors from each subspace results in a vanishing sum.

Discussion

Let $\mathcal{A}, \mathcal{B}$ be subspaces of a vector space.

Cartesian Product of Vector Spaces

Let $\mathcal{V}_1 ⋯ \mathcal{V}_n$ be vector spaces over the same field. Then the cartesian product of these spaces is defined as the set of all lists whose indexed elements are drawn from their corresponding vector spaces:

Discussion

• The cartesian product of vector spaces is always a vector space.
• $𝖥^n \times 𝖥^m$ is isomorphic to $𝖥^{n + m}$.

Let $(\vec{v}_1 ⋯ \vec{v}_n)$ be a basis for the vector space $\mathcal{V}$ over a field $𝖥$. Then the span of $\operatorname{span}(\vec{v}_1 ⋯ \vec{v}_n)$ can be alternatively described as:

If we consider every subspace $\mathcal{V}_i$ as $\{ \vec{v}_i \} \times 𝖥$, then this is equivalent to the cartesian product of subspaces $∏ \mathcal{V}_i$, and the cardinality of this set is the cardinality of $\mathcal{V}$.

Affine Subspace

Let $\mathcal{V}_1$ be a subset of the vector space $\mathcal{V}$. Then an affine subset can be defined:

The affine subset $\vec{v} + \mathcal{V}_1$ is defined as parallel to $\mathcal{V}_1$.

• The resulting set is not a subspace due to missing $\vec{0}$.
• Two subsets which are parallel to $\mathcal{V}_1$ are themselves either parallel or equal (?).

Quotient Space

Let $\mathcal{V}_1$ be a linear subspace of $\mathcal{V}$. Then the quotient space $\mathcal{V} / \mathcal{V}_1$ is the set of all affine subsets of $\mathcal{V}$ which are parallel to $\mathcal{V}_1$.

The dimension of a quotient space is

Addition and Scalar Multiplication of Quotient Spaces

Where $\sigma$ is a scalar from the common field.

Quotient Map

Let $\mathcal{V}_1$ be a linear subspace of $\mathcal{V}$. The quotient map $π : \mathcal{V} → \mathcal{V} / \mathcal{V}_1$ is defined:

Where $\vec{v} ∈ \mathcal{V}$.

Commentary

• All these properties and definitions depend only on vector spaces. No inner products or dot products.