User:SirMeowMeow/sandbox/Vector Space

Definition

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 this space is a vector.

To be compatible with vector scaling, a vector space must be accompanied with a field (or ring for generality), typically denoted ${\mathsf {F}}$ or ${\mathsf {K}}$. 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 the identity element under vector vector addition, and is known as the trivial vector space.

Abstract Definition

A vector space ${\mathcal {V}}$ over a field ${\mathsf {F}}$ is a set defined with an abelian addition $+:{\mathcal {V}}\times {\mathcal {V}}\to {\mathcal {V}}$ and a vector scaling function ${\mathsf {F}}\times {\mathcal {V}}\to {\mathcal {V}}$ of the form $(\alpha ,{\vec {x}})\mapsto \alpha {\vec {x}}$, such that:

Vector Addition

${\vec {x}}+{\vec {0}}={\vec {x}}$	Identity element
${\vec {x}}+{\vec {y}}={\vec {y}}+{\vec {x}}$	Commutative
${\vec {x}}+({\vec {y}}+{\vec {z}})=({\vec {x}}+{\vec {y}})+{\vec {z}}$	Associative
${\vec {x}}+(-{\vec {x}})={\vec {0}}$	Inverse element

Where ${\vec {0}},{\vec {x}},{\vec {y}},{\vec {z}}\in {\mathcal {V}}$.

Vector Scaling

$1{\vec {x}}={\vec {x}}$	Identity element
$\alpha (\beta {\vec {x}})=(\alpha \beta ){\vec {x}}$	Vector scaling is compatible with field multiplication
$(\alpha +\beta ){\vec {x}}=\alpha {\vec {x}}+\beta {\vec {x}}$	Field addition distributes over vector scaling
$\alpha ({\vec {x}}+{\vec {y}})=\alpha {\vec {x}}+\alpha {\vec {y}}$	Vector scaling distributes over field addition

Where $1,\alpha ,\beta \in {\mathsf {F}}$.

Finite Definition

A finite list of $n$ scalars from ${\mathsf {F}}$ is written ${\mathsf {F}}^{n}$, and all finite-dimensional vector spaces are isomorphic or linearly equivalent to the vector space defined over ${\mathsf {F}}^{n}$.

Vector Addition

Let $[x_{1}\cdots x_{n}],[\ y_{1}\cdots y_{n}]\in {\mathsf {F}}^{n}$. Finite vector addition is defined as pointwise scalar addition.

${\begin{bmatrix}x_{1}\\\vdots \\x_{n}\\\end{bmatrix}}+{\begin{bmatrix}y_{1}\\\vdots \\y_{n}\\\end{bmatrix}}:={\begin{bmatrix}x_{1}+y_{1}\\\vdots \\x_{n}+y_{n}\\\end{bmatrix}}$

Vector Scaling

Let $\sigma \in {\mathsf {F}}$ and $[x_{1}\cdots x_{n}]\in {\mathsf {F}}^{n}$. Finite vector scaling is defined as pointwise scalar multiplication.

$\sigma {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\\\end{bmatrix}}:={\begin{bmatrix}\sigma x_{1}\\\vdots \\\sigma x_{n}\\\end{bmatrix}}$

Examples of Vector Spaces

Functional Spaces

The vector space of polynomials forms an infinite-dimensional vector space.

Bitvectors

The field of two elements $\{0,1\}$ is considered the smallest field, and is named the Galois field of two elements, denoted as $\operatorname {GF} (2)$ or ${\mathsf {F}}_{2}$. Because this vector space does not admit any inner product, the dot product of two identical vectors from ${\mathsf {F}}_{2}^{n}$ may be zero.

${\begin{bmatrix}1\\1\\\end{bmatrix}}\cdot {\begin{bmatrix}1\\1\\\end{bmatrix}}=(1)(1)+(1)(1)=0$

Finite Vector Spaces

Finite vector spaces may be defined only with finite groups over finite fields with a prime order of elements, denoted ${\mathsf {F}}_{p}^{n}$ for any prime $p$.

Polynomial F[x]

The set of polynomial terms below spans ${\mathsf {F}}[x]$, and is countably infinite.

$\{x^{n}\mid n\in \mathbb {N} \}$

The polynomials form a vector space because elements from ${\mathsf {F}}[x]$ form a commutative group, and because polynomials are closed under vector scaling.

A set polynomials are independent when trivial combination is the only zero polynomial.

The derivative is a linear endomorphism which is defined for all polynomials, and it's surjective but not injective. But for endomorphisms on finitely generated modules, surjectivity, injectivity, and isomorphism are all equivalent conditions. Thus ${\mathsf {F}}[x]$ forms an infinite-dimensional vector space.

Linear Combination

Main article: Linear combination

Let ${\mathcal {V}}=\{{\vec {v}}_{1}\cdots {\vec {v}}_{n}\}$ be a subset of a vector space, and let $\sigma _{1}\cdots \sigma _{n}\in {\mathsf {F}}$. Then a linear combination of ${\mathcal {V}}$ is defined as any vector which is the sum of scaled vectors in ${\mathcal {V}}$.

$\sum _{i=1}^{n}\sigma _{i}{\vec {v}}_{i}$

A trivial combination means every $\sigma _{i}=0$.

Span

Main article: Linear span

Let ${\mathcal {V}}=\{{\vec {v}}_{1}\cdots {\vec {v}}_{n}\}$ be a subset of a vector space, and let $\sigma _{1}\cdots \sigma _{n}\in {\mathsf {F}}$. Then the span of ${\mathcal {V}}$ is defined as the set of all combinations from ${\mathcal {V}}$.

$\operatorname {span} {\mathcal {V}}:=\left\{\sum _{i=1}^{n}\sigma _{i}{\vec {v}}_{i}\right\}$

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.

Independence

Main article: Linear independence

Let ${\mathcal {V}}=\{{\vec {v}}_{1}\cdots {\vec {v}}_{n}\}$ be a subset of a vector space, and let $\sigma _{1}\cdots \sigma _{n}\in {\mathsf {F}}$. Then ${\mathcal {V}}$ is independent iff the trivial combination is the only vanishing sum.

$\sum _{i=1}^{n}\sigma _{i}{\vec {v}}_{i}={\vec {0}}$

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

Observations

• ${\mathcal {V}}$ 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

Main articles: Basis (linear algebra) and Dimension (vector space)

The basis of a vector space ${\mathcal {V}}$ is any independent set whose span is exactly ${\mathcal {V}}$, and the elements of a basis are called basis vectors. The dimension of ${\mathcal {V}}$ is the number of vectors or cardinality of its basis, written as $\dim({\mathcal {V}})$.

If the basis of ${\mathcal {V}}$ has finite cardinality, then ${\mathcal {V}}$ is defined as a finite-dimensional vector space; otherwise, ${\mathcal {V}}$ is an infinite-dimensional vector space.

Observations

• Every vector space has a basis.
• All choices of basis for a vector space will have the same cardinality.
• Any independent subset of ${\mathcal {V}}$ which isn't spanning can be expanded into a basis.
• Any dependent subset which spans ${\mathcal {V}}$ can remove vectors until it is a basis.

Linear Subspace

Main article: Linear subspace

A linear subspace of a vector space ${\mathcal {V}}$ is any subset which is also a vector space under the same abelian addition and vector scaling as ${\mathcal {V}}$. The trivial vector space is the smallest subspace which contains only the identity element of vector addition.

The subspaces ${\mathcal {V}}_{1}\cdots {\mathcal {V}}_{n}$ are independent iff for any pair of subspaces the only vector in common is ${\vec {0}}$. Thus for any pair of independent subspaces their intersection is the trivial subspace.

• ${\mathcal {V}}_{1}\cdots {\mathcal {V}}_{n}$ is independent iff the trivial combination is the unique vanishing sum.
• ${\mathcal {V}}_{1}\cdots {\mathcal {V}}_{n}$ is independent iff every subspace contributes to the span of the subspace sum.

Construction of Vector Spaces

Sum of Subspaces

The sum of subspaces ${\mathcal {V}}_{1}\cdots {\mathcal {V}}_{n}$ is the set of all vector sums with summands drawn from each corresponding subspace.

${\mathcal {V}}_{1}+\cdots +{\mathcal {V}}_{n}:=\{{\vec {v}}_{1}+\cdots +{\vec {v}}_{n}\mid {\vec {v}}_{i}\in {\mathcal {V}}_{i}\}$

Direct Sum of Subspaces

A list of subspaces ${\mathcal {V}}_{1}\cdots {\mathcal {V}}_{n}\subseteq {\mathcal {V}}$ is independent if for any pair of subspaces the only vector in common is ${\vec {0}}$. Thus for any pair of independent subspaces their intersection is the trivial subspace.

The direct sum of subspaces ${\mathcal {V}}_{1}\cdots {\mathcal {V}}_{n}$ is sum of independent subspaces, and is written:

${\mathcal {V}}_{1}\oplus \cdots \oplus {\mathcal {V}}_{n}$

• $\dim({\mathcal {V}}_{1}+{\mathcal {V}}_{2})=\dim({\mathcal {V}}_{1})+\dim({\mathcal {V}}_{2})-\dim({\mathcal {V}}_{1}\cap {\mathcal {V}}_{2})$
• $\dim({\mathcal {V}}_{1}\oplus {\mathcal {V}}_{2})=\dim({\mathcal {V}}_{1})+\dim({\mathcal {V}}_{2})$

Cartesian Product of Subspaces

Let ${\mathcal {V}}_{1}\cdots {\mathcal {V}}_{n}$ 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:

${\mathcal {V}}_{1}\times \cdots \times {\mathcal {V}}_{n}:=\{({\vec {v}}_{1}\cdots {\vec {v}}_{n})\mid {\vec {v}}_{i}\in {\mathcal {V}}_{i}\}$

• $\dim({\mathcal {V}}_{1}\times \cdots \times {\mathcal {V}}_{n})=\dim {\mathcal {V}}_{1}+\cdots +\dim {\mathcal {V}}_{n}$

Maps on Vector Spaces

Main articles: Linear map and Homomorphism

A mapping between vector spaces which preserves vector addition and scaling may be known as a linear map, operator, homomorphism, or function.

Let ${\mathcal {V}},{\mathcal {W}}$ be vector spaces over a ring or field ${\mathsf {F}}$. Then a linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ is defined as any mapping such that:

$\Phi (\sigma _{1}{\vec {v}}_{1}+\cdots +\sigma _{n}{\vec {v}}_{n})=\sigma _{1}\Phi ({\vec {v}}_{1})+\cdots +\sigma _{n}\Phi ({\vec {v}}_{n})$

Where $\sigma _{i}\in {\mathsf {F}}$, and ${\vec {v}}_{i}\in {\mathcal {V}}$, and ${\vec {w}}_{i}\in {\mathcal {W}}$.

Observations

• Linear combinations in ${\mathcal {V}}$ are mapped to linear combinations in ${\mathcal {W}}$.
• Group homomorphism implies identity is mapped to identity, and inverses are mapped to inverses: ${\vec {v}}+(-{\vec {v}})\mapsto {\vec {w}}+(-{\vec {w}})$.

Inner Product Spaces

Main article: Inner product space

Let ${\mathcal {V}}$ be a vector space over a field ${\mathsf {F}}$. An inner product is a mapping $\langle \cdot ,\cdot \rangle :{\mathcal {V}}\times {\mathcal {V}}\to {\mathsf {F}}$ such that:

$\sigma \langle {\vec {x}}_{1}+{\vec {x}}_{2},{\vec {y}}\rangle =\langle \sigma {\vec {x}}_{1},{\vec {y}}\rangle +\langle \sigma {\vec {x}}_{2},{\vec {y}}\rangle$	Linearity in the first argument
$\langle {\vec {x}},{\vec {y}}\rangle =\langle {\overline {{\vec {y}},{\vec {x}}}}\rangle$	Conjugate symmetry
$0<\langle {\vec {x}},{\vec {y}}\rangle$	Positive-definite

A vector space for which an inner product can be defined is called an inner product space.

Examples

Dot Product

Let ${\vec {x}}=[x_{1}\cdots x_{n}],\ {\vec {y}}=[y_{1}\cdots y_{n}]$ be from the vector space ${\mathcal {V}}$ over a field ${\mathsf {F}}$. Then dot product is a map ${\mathcal {V}}\times {\mathcal {V}}\to {\mathsf {F}}$ such that:

${\vec {x}}\cdot {\vec {y}}:=\sum _{i=1}^{n}x_{i}y_{i}$

For vector spaces where the dot product qualifies as an inner product, the dot product is known as a definition for Euclidean distance.

Affine Subset

Main articles: Affine subspace and Quotient space (linear algebra)

Let ${\mathcal {V}}_{1}$ be a subset of the vector space ${\mathcal {V}}$. Then an affine subset can be defined as the set:

${\vec {v}}+{\mathcal {V}}_{1}:=\{{\vec {v}}+{\vec {v}}_{1}\mid {\vec {v}}\in {\mathcal {V}},{\vec {v}}_{1}\in {\mathcal {V}}_{1}\}$

Any affine subset ${\vec {v}}+{\mathcal {V}}_{1}$ is defined as parallel to ${\mathcal {V}}_{1}$.

Vector Space of Affine Subsets

Let ${\mathcal {V}}_{1}$ be a subset of the vector space ${\mathcal {V}}$, and let ${\vec {v}},{\vec {w}}\in {\mathcal {V}}$.

$({\vec {v}}+{\mathcal {V}}_{1})+({\vec {w}}+{\mathcal {V}}_{1}):=({\vec {v}}+{\vec {w}})+{\mathcal {V}}_{1}$

$\sigma ({\vec {v}}+{\mathcal {V}}_{1}):=\sigma {\vec {v}}+{\mathcal {V}}_{1}$

Quotient Space

The quotient space ${\mathcal {V}}/{\mathcal {V}}_{1}$ is defined as the set of affine subsets which are parallel to ${\mathcal {V}}_{1}$.

${\mathcal {V}}/{\mathcal {V}}_{1}:=\{{\vec {v}}+{\mathcal {V}}_{1}\mid {\vec {v}}\in {\mathcal {V}}\}$

• $\dim {\mathcal {V}}/{\mathcal {V}}_{1}=\dim {\mathcal {V}}-\dim {\mathcal {V}}_{1}$

Quotient Map

Let ${\mathcal {V}}_{1}$ be a linear subspace of ${\mathcal {V}}$. The quotient map $\pi :{\mathcal {V}}\to {\mathcal {V}}/{\mathcal {V}}_{1}$ is defined:

$\pi ({\vec {v}}):={\vec {v}}+{\mathcal {V}}_{1}$

Sources

Textbook

• Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0.
• Gallian, Joseph A. (2012). Contemporary Abstract Algebra (8th ed.). Cengage. ISBN 978-1-133-59970-8.
• Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces (2nd ed.). Springer. ISBN 0-387-90093-4.
• Hefferon, Jim (2020). Linear Algebra (4th ed.). Orthogonal Publishing. ISBN 978-1-944325-11-4.
• Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9.
• Roman, Steven (2005). Advanced Linear Algebra (2nd ed.). Springer. ISBN 0-387-24766-1.
• Strang, Gilbert (2016). Introduction to Linear Algebra (5th ed.). Wellesley Cambridge Press. ISBN 978-0-9802327-7-6.

Web

• "Definition: Vector Space". ProofWiki. 7 Feb 2021. Retrieved 17 Feb 2021.{{cite web}}: CS1 maint: url-status (link)
• "Vector Space". nLab. 3 Nov 2020. Retrieved 17 Feb 2021.{{cite web}}: CS1 maint: url-status (link)
• "Lemma 10.16.4". Stacks Project. 21 Mar 2015. Retrieved 13 Mar 2021.{{cite web}}: CS1 maint: url-status (link)