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:

Where $$\sigma_i \in \mathsf{F}$$, and $$\vec{v}_i \in \mathcal{V}$$, and $$\vec{w}_i \in \mathcal{W}$$.

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 $$\mathcal{V}$$.

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

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:

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 $$\vec{0}$$.

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

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.

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

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

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

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

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.

Epimorphism
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 $$\mathcal{V}$$.

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

Monomorphism
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.

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

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.

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
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.

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:

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 $$\mathcal{V}^*$$ 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 $$\mathcal{V}_1$$.

Textbooks














Web






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