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

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 ${\mathcal {V}}$ .^[1]^[2]^[a]

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

The rank of a map is the dimension of its image.^[3]^[4]^[5]^[b]

\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

The kernel or nullspace of a linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ is the subset of ${\mathcal {V}}$ which is mapped to ${\vec {0}}$ .^[6]^[7]

\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.^[c]

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

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.^[8]^[9]^[10]^[d]

\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 $\hom _{\mathsf {K}}({\mathcal {V,W}})$ ^[11]^[12] or ${\mathcal {L(V,W)}}$ .^[13]^[14]^[15]

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^[16]

(\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^[17]^[18]

(\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.^[19]

$\Omega (\Psi \Phi )=(\Omega \Psi )\Phi$ 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

Homomorphism

A homomorphism is a structure-preserving map between two algebraic objects of the same type.^[e] 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

An epimorphism is a right-cancellative^[f] 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

A monomorphism is a left-cancellative^[f] 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

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.^[g]

Equivalent conditions

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

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

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

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 ${\mathcal {V}}^{*}$ ^[20] or ${\mathcal {V}}'$ .^[21]^[22] 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

Notes

^ Alternative notation for image includes ${\mathfrak {R}}$ from Halmos (1974) p. 88, § 49.
^ Alternative notation for rank includes $\rho$ from Katznelson & Katznelson (2008), p. 52, §2.5.1 and Halmos (1974), p. 90, § 50.
^ Alternative notation for nullity includes $\nu$ from Katznelson & Katznelson (2008), p. 52, § 2.5.1 and Halmos (1974), p. 90, § 50.
^ Alternative notation for kernel includes ${\mathfrak {N}}$ from Halmos (1974) p. 88, § 49.
^ Let $({\mathcal {V}},+)$ and $({\mathcal {W}},\times )$ be magmas. Then a homomorphism $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ is a map such that: $\Phi ({\vec {v}}_{1}+{\vec {v}}_{2})=\Phi ({\vec {v}}_{1})\times \Phi ({\vec {v}}_{2})$ The definition can be extended to monoids or semigroups by preserving the identity element $0$ . $0_{\mathcal {V}}\operatorname {\stackrel {\Phi }{\longmapsto }} 0_{\mathcal {W}}$
^ ^a ^b In the context of commutative groups there is no difference between cancellation and the existence of inverses.
^ This property doesn't hold for infinite-dimensional spaces. As a counterexample, polynomials ${\mathsf {F}}[x]$ have an infinite-dimensional basis, and the derivative is a surjective endomorphism on ${\mathsf {F}}[x]$ , but the derivative is not injective.

Citations

^ Axler (2015) p. 61, § 3.17
^ Katznelson & Katznelson (2008) p. 52 § 2.5.1
^ Hefferon (2020) p. 200, ch. 3, Definition 2.1
^ Valenza (1993) p. 71, § 4.3
^ Halmos (1974) p. 90, § 50
^ Axler (2015) p. 59, § 3.12
^ Katznelson & Katznelson (2008) p. 51 § 2.5.1
^ Axler (2015) p. 63, § 3.22
^ Katznelson & Katznelson (2008) p. 52, § 2.5.1
^ Valenza (1993) p. 71, § 4.3
^ Tu (2011) p. 19, § 3.1
^ Valenza (1993) p. 100, § 6.1
^ Axler (2015) p. 52, § 3.3
^ Katznelson & Katznelson (2008) p. 39 § 2.2.1
^ Roman (2005) p. 55, ch. 2
^ Axler (2015) p. 55, § 3.6
^ Axler (2015) p. 55, § 3.8
^ Katznelson & Katznelson (2008) p. 39, § 2.2.1
^ Axler (2015) p. 56, § 3.9
^ Katznelson & Katznelson (2008) p. 37, §2.1.3
^ Axler (2015) p. 101, §3.94
^ Halmos (1974) p. 20, §13

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)

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

[3] Alternative notation for image includes ${\mathfrak {R}}$ from Halmos (1974) p. 88, § 49.

[7] Alternative notation for rank includes $\rho$ from Katznelson & Katznelson (2008), p. 52, §2.5.1 and Halmos (1974), p. 90, § 50.

[10] Alternative notation for nullity includes $\nu$ from Katznelson & Katznelson (2008), p. 52, § 2.5.1 and Halmos (1974), p. 90, § 50.

[14] Alternative notation for kernel includes ${\mathfrak {N}}$ from Halmos (1974) p. 88, § 49.

[24] Let $({\mathcal {V}},+)$ and $({\mathcal {W}},\times )$ be magmas. Then a homomorphism $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ is a map such that: $\Phi ({\vec {v}}_{1}+{\vec {v}}_{2})=\Phi ({\vec {v}}_{1})\times \Phi ({\vec {v}}_{2})$ The definition can be extended to monoids or semigroups by preserving the identity element $0$ . $0_{\mathcal {V}}\operatorname {\stackrel {\Phi }{\longmapsto }} 0_{\mathcal {W}}$

[:0-25] In the context of commutative groups there is no difference between cancellation and the existence of inverses.

[26] This property doesn't hold for infinite-dimensional spaces. As a counterexample, polynomials ${\mathsf {F}}[x]$ have an infinite-dimensional basis, and the derivative is a surjective endomorphism on ${\mathsf {F}}[x]$ , but the derivative is not injective.

[1] Axler (2015) p. 61, § 3.17

[2] Katznelson & Katznelson (2008) p. 52 § 2.5.1

[4] Hefferon (2020) p. 200, ch. 3, Definition 2.1

[5] Valenza (1993) p. 71, § 4.3

[6] Halmos (1974) p. 90, § 50

[8] Axler (2015) p. 59, § 3.12

[9] Katznelson & Katznelson (2008) p. 51 § 2.5.1

[11] Axler (2015) p. 63, § 3.22

[12] Katznelson & Katznelson (2008) p. 52, § 2.5.1

[13] Valenza (1993) p. 71, § 4.3

[15] Tu (2011) p. 19, § 3.1

[16] Valenza (1993) p. 100, § 6.1

[17] Axler (2015) p. 52, § 3.3

[18] Katznelson & Katznelson (2008) p. 39 § 2.2.1

[19] Roman (2005) p. 55, ch. 2

[20] Axler (2015) p. 55, § 3.6

[21] Axler (2015) p. 55, § 3.8

[22] Katznelson & Katznelson (2008) p. 39, § 2.2.1

[23] Axler (2015) p. 56, § 3.9

[27] Katznelson & Katznelson (2008) p. 37, §2.1.3

[28] Axler (2015) p. 101, §3.94

[29] Halmos (1974) p. 20, §13

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v t e Linear algebra
Outline Glossary
Basic concepts	Scalar Vector Vector space Scalar multiplication Vector projection Linear span Linear map Linear projection Linear independence Linear combination Multilinear map Basis Change of basis Row and column vectors Row and column spaces Kernel Eigenvalues and eigenvectors Transpose Linear equations
Matrices	Block Decomposition Invertible Minor Multiplication Rank Transformation Cramer's rule Gaussian elimination Productive matrix
Bilinear	Orthogonality Dot product Hadamard product Inner product space Outer product Kronecker product Gram–Schmidt process
Multilinear algebra	Determinant Cross product Triple product Seven-dimensional cross product Geometric algebra Exterior algebra Bivector Multivector Tensor Outermorphism
Vector space constructions	Dual Direct sum Function space Quotient Subspace Tensor product
Numerical	Floating-point Numerical stability Basic Linear Algebra Subprograms Sparse matrix Comparison of linear algebra libraries
Category

Definition

Image and Kernel of a linear map

Image and Rank

Kernel and Nullity

Rank-nullity theorem

Vector space of linear maps

Sum and scalar multiplication of linear maps

Discussion

Product of linear maps

Ring of linear maps

Composition of linear maps

Structure Preserving Maps

Homomorphism

Epimorphism

Monomorphism

Isomorphism

Equivalent conditions

Endomorphism

Discussion

Automorphism

Observations

Identity Map

Invertible Maps

Linear Forms

Transposition

Subspace Restriction of Linear Map

Famous Commentary

Notes

Citations

Sources

Textbooks

Web

Related