User:SirMeowMeow/sandbox/Linear Maps

Definition

A mapping between vector spaces which preserves abelian addition and vector 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: ${\vec {v}}+(-{\vec {v}})\mapsto {\vec {w}}+(-{\vec {w}})$.

Image and Kernel of a linear map

Image and Rank

Main articles: Image (mathematics) and Rank (linear algebra)

The image or range of a linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ is the subset of ${\mathcal {W}}$ which has a mapping from ${\mathcal {V}}$.

$\operatorname {img} \Phi :=\{\Phi {\vec {v}}\mid {\vec {v}}\in {\mathcal {V}}\}$

The rank of a map is the dimension of its image.

$\operatorname {rank} \Phi :=\dim \operatorname {img} \Phi$

Any injective linear map (monomorphism) is known as full-rank, and otherwise known as rank-deficient. For linear maps between finite-dimensional vector spaces, rank-deficiency may be defined:

$\dim {\mathcal {W}}-\operatorname {rank} \Phi$

Kernel and Nullity

Main article: Kernel (linear algebra)

The kernel or nullspace of a linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ is the subset of ${\mathcal {V}}$ which is mapped to ${\vec {0}}$.

$\ker \Phi :=\{{\vec {v}}\in {\mathcal {V}}\mid \Phi {\vec {v}}={\vec {0}}\}$

The nullity of a linear map is the dimension of its nullspace.

$\operatorname {null} \Phi :=\dim \ker \Phi$

Rank-nullity theorem

Main article: Rank–nullity theorem

The rank-nullity theorem states that for any linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ whose domain is finite-dimensional, the dimension of ${\mathcal {V}}$ equals the sum of the map's rank and nullity.

$\dim \operatorname {img} \Phi +\dim \ker \Phi =\dim {\mathcal {V}}$

$\operatorname {rank} \Phi +\operatorname {null} \Phi =\dim {\mathcal {V}}$

Vector space of linear maps

Let ${\mathcal {V,W}}$ be vector spaces over a field ${\mathsf {K}}$ (or division ring for generality). The set of all linear maps from ${\mathcal {V}}$ to ${\mathcal {W}}$ may be denoted $\hom _{\mathsf {K}}({\mathcal {V,W}})$ or ${\mathcal {L(V,W)}}$.

Sum and scalar multiplication of linear maps

Let $\Phi ,\Psi :{\mathcal {V\to W}}$ be a ${\mathsf {K}}$-linear map. The sum and scalar multiplication of linear maps is defined

$(\Phi +\Psi ){\vec {v}}:=\Phi {\vec {v}}+\Psi {\vec {v}}$

$\sigma _{1}(\sigma _{2}\Phi ){\vec {v}}:=(\sigma _{1}\sigma _{2})\Phi {\vec {v}}$

where ${\vec {v}}\in {\mathcal {V}}$ and $\sigma _{1},\sigma _{2}\in {\mathsf {K}}$.

Discussion

• Under addition and scalar multiplication the set of linear maps ${\mathcal {L(V,W)}}$ forms a vector space over ${\mathsf {K}}$.
• For finite-dimensional vector spaces, $\dim {\mathcal {L(V,W)}}=(\dim {\mathcal {V}})(\dim {\mathcal {W}})$.

Product of linear maps

Let $\Phi _{i}:{\mathcal {V}}\to {\mathcal {W}}$, and $\Psi _{i}:{\mathcal {W}}\to {\mathcal {X}}$, and $\Omega _{i}:{\mathcal {X}}\to {\mathcal {Y}}$ be linear maps. The multiplication or product of linear maps is defined

$(\Psi \Phi ){\vec {v}}:=\Psi (\Phi {\vec {v}})$

where ${\vec {v}}\in {\mathcal {V}}$.

Ring of linear maps

The vector space of linear maps forms a ring under addition and scalar multiplication.

$\Omega (\Psi \Phi )=(\Omega \Psi )\Phi$	Associative
$\Phi \mathrm {I} _{\mathcal {V}}=\mathrm {I} _{\mathcal {W}}\Phi =\Phi$	Unique right and left identity
$\Psi (\Phi _{1}+\Phi _{2})=\Psi \Phi _{1}+\Psi \Phi _{2}$	Right distributive
$(\Psi _{1}+\Psi _{2})\Phi =\Psi _{1}\Phi +\Psi _{2}\Phi$	Left distributive

Composition of linear maps

The composition of linear maps is inherited from the composition of functions

$\Psi \circ \Phi :=\Psi (\Phi {\vec {v}})$

where ${\vec {v}}\in {\mathcal {V}}$.

Structure Preserving Maps

A conceptual dependency chart from automorphism to homomorphism.

Homomorphism

Main article: Homomorphism

A homomorphism is a structure-preserving map between two algebraic objects of the same type. Linear homomorphisms are maps between vector spaces which preserves vector addition and scalar multiplication — this is an equivalent definition for linear maps.

The set of all linear maps ${\mathcal {V}}\to {\mathcal {W}}$ over a field ${\mathsf {K}}$ may be denoted $\hom _{\mathsf {K}}({\mathcal {V,W}})$.

Every vector space has an underlying commutative group, and a group homomorphism will map identity to identity, and inverses to inverses.

$({\vec {v}}+-{\vec {v}})_{\mathcal {V}}\longmapsto ({\vec {w}}+-{\vec {w}})_{\mathcal {W}}$

Epimorphism

Main articles: Epimorphism and Surjection

An epimorphism is a right-cancellative morphism, and a linear epimorphism is a linear surjection. A linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ is surjective when every element in ${\mathcal {W}}$ has a mapping from ${\mathcal {V}}$.

$\forall ({\vec {w}}\in {\mathcal {W}}),\ \exists ({\vec {v}}\in {\mathcal {V}}):\Phi {\vec {v}}={\vec {w}}$

Every linear surjection has a right inverse $\Phi _{\textrm {R}}^{-1}:{\mathcal {W}}\to {\mathcal {V}}$ such that:

$\Phi \circ \Phi _{\textrm {R}}^{-1}=\mathrm {I} _{\mathcal {W}}$

$\Phi _{\textrm {R}}^{-1}=\Phi ^{\intercal }(\Phi \Phi ^{\intercal })^{-1}$

Monomorphism

Main articles: Monomorphism and Injection (mathematics)

A monomorphism is a left-cancellative morphism, and a linear monomorphism is an injective (or one-to-one) linear map. A linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ is injective when every domain element is uniquely mapped to every image element.

$\Phi {\vec {v}}_{1}=\Phi {\vec {v}}_{2}\longrightarrow {\vec {v}}_{1}={\vec {v}}_{2}$

Every linear injection has a left inverse $\Phi _{\textrm {L}}^{-1}:{\mathcal {W}}\to {\mathcal {V}}$ such that:

$\Phi _{\textrm {L}}^{-1}\circ \Phi =\mathrm {I} _{\mathcal {V}}$

$\Phi _{\textrm {L}}^{-1}=(\Phi ^{\intercal }\Phi )^{-1}\Phi ^{\intercal }$

Isomorphism

Main articles: Isomorphism and Bijection

A linear isomorphism is any bijective linear map. This means that any isomorphism will have the same left and right inverse, as well as the same left and right identity.

$(\Phi )(\Phi ^{-1})=(\Phi ^{-1})(\Phi )=\mathrm {I}$

For finite-dimensional modules, linear surjection, injection, and bijection are all equivalent conditions.

Equivalent conditions

The following is a list of properties which are equivalent to linear isomorphism.

Endomorphism

Main article: Endomorphism

A linear endomorphism is a linear map $\Phi :{\mathcal {V}}\to {\mathcal {V}}$ with the same domain and codomain, and the set of all endomorphisms on ${\mathcal {V}}$ may be denoted ${\mathcal {L}}({\mathcal {V}})$.

Discussion

• One example of a linear endomorphism on any vector space is the zero map $\Phi {\vec {v}}\mapsto {\vec {0}}$.

• For endomorphisms over finitely generated modules (thus vector spaces), surjectivity, injectivity, and bijectivity are all equivalent.

Automorphism

Main articles: Automorphism and Permutations

An automorphism is an isomorphic endomorphism, and all automorphisms are also permutations.

The finite symmetric group $\mathrm {S} _{n}$ is the set of all automorphisms on any finite set. The set of all automorphisms on ${\mathcal {V}}$ may be known as the general linear group of ${\mathcal {V}}$, denoted $\operatorname {GL} ({\mathcal {V}})$.

Observations

• An automorphism that always exists for any vector space is the identity map.
• An automorphism is slightly different than an isomorphism, even though they will both be described by square matrices; for example, there is an isomorphism from the reals ${\mathsf {R}}^{2}$ to the complex numbers ${\mathsf {C}}$, but this is not an automorphism.

Identity Map

For any linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ there also exists linear maps $\mathrm {I} _{\mathcal {V}},\mathrm {I} _{\mathcal {W}}$ which act as the unique right and left identity element under the product of maps.

$(\Phi )(\mathrm {I} _{\mathcal {V}})=(\mathrm {I} _{\mathcal {W}})(\Phi )=\Phi$

Any linear map which fulfills this condition is known as the identity map, denoted $\mathrm {I}$, or with a subscript $\mathrm {I} _{n}$ for some dimension $n$.

Invertible Maps

A linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ is invertible if there exists a linear map $\Phi ^{-1}:{\mathcal {W}}\to {\mathcal {V}}$ such that:

$(\Phi ^{-1})(\Phi )=\mathrm {I} _{\mathcal {V}}$

$(\Phi )(\Phi ^{-1})=\mathrm {I} _{\mathcal {W}}$

Linear invertibility, bijectivity, and isomorphism are all equivalent terms. With linear endomorphisms between finite-dimensional modules, surjectivity, injectivity, and bijectivity are all equivalent conditions.

Linear Forms

Main articles: Linear form and Dual space

Let ${\mathcal {V}}$ be a vector space over ${\mathsf {F}}$. Then a linear form is any linear map ${\mathcal {V}}\to {\mathsf {F}}$.

The algebraic dual space is the set of all linear forms on ${\mathcal {V}}$, and is denoted ${\mathcal {V}}^{*}$ or ${\mathcal {V}}'$. If the vector space has a defined topology, then it may be known as a topological dual space.

A linear map ${\mathcal {V}}\times {\mathcal {V}}^{*}\to {\mathsf {F}}$ is known as a natural pairing.

Transposition

Let $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ be a linear map. Then there exists a dual map $\Phi ^{*}:{\mathcal {V}}^{*}\to {\mathcal {W}}^{*}$ known as the transposition.

Subspace Restriction of Linear Map

Let $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ be a linear map, and let ${\mathcal {V}}_{1}$ be a subspace of ${\mathcal {V}}$. Then $\Phi _{{\mathcal {V}}_{1}}$ denotes the restriction of $\Phi$ to act only on the subspace of ${\mathcal {V}}_{1}$.

Famous Commentary

There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.

— Jean Dieudonné, Treatise on Analysis, Volume 1

We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.

— Irving Kaplansky, in writing about Paul Halmos

Sources

Textbooks

• Axler, Sheldon Jay (2015). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 978-3-319-11079-0.

• Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces. Undergraduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-90093-4.

• Hefferon, Jim (2020). Linear Algebra (4th ed.). 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. Undergraduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-24766-1.

• Tu, Loring W. (2011). An Introduction to Manifolds. Universitext (2nd ed.). Springer. ISBN 978-0-8218-4419-9.

• Valenza, Robert J. (1993) [1951]. Linear Algebra: An Introduction to Abstract Mathematics. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 3-540-94099-5.

Web

• "Linear Algebra". Stanford Encyclopedia of Philosophy. Retrieved 6 Feb 2021.{{cite web}}: CS1 maint: url-status (link)

• "Linear Map". nLab. Retrieved 6 Feb 2021.{{cite web}}: CS1 maint: url-status (link)

• "Vector Space". nLab. Retrieved 6 Feb 2021.{{cite web}}: CS1 maint: url-status (link)

Related

Category:Abstract algebra Category:Functions and mappings Category:Linear algebra Category:Transformation (function)