Linear Maps

Let $\mathcal{V}, \mathcal{W}$ be vector spaces over a field $𝖥$. Then a linear map $\Phi: \mathcal{V} → \mathcal{W}$ is a function which preserves the vector space properties of $\mathcal{V}$ when mapping to $\mathcal{W}$.

Where $σ_i ∈ 𝖥$ and $\vec{v}_i ∈ \mathcal{V}$.

A linear map may also be known as a linear function, operator, transformation, or homomorphism.

Discussion

• Group addition is preserved.
• Vector scaling is homogeneous.
• Combinations in $\mathcal{V}$ are mapped to combinations in $\mathcal{W}$.
• $\mathcal{V}, \mathcal{W}$ may have different underlying fields.

Image and Rank

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

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

Any linear map which is injective is full-rank or monomorphic, and otherwise the map is considered rank-deficient. In the finite-dimensional case, rank-deficiency can be defined as the dimension of the domain subtracted by the rank.

Nullspace and Nullity

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

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

Discussion

• The nullspace is a vector space.
• Group homomorphism means $(\vec{v} + -\vec{v})_\mathcal{V} ↦ (\vec{w} + -\vec{w})_\mathcal{W}$.

Rank Nullity Theorem

Let $\Phi: \mathcal{V} → \mathcal{W}$ be a linear map with finite-dimensional $\mathcal{V}$. The rank-nullity theorem states:

The rank-nullity theorem can be modified for infinite-dimensional spaces by interpreting the sum of disjoint cardinalities as the cardinality of their union.

Vector Space of Linear Maps

The set of all linear maps from $\mathcal{V}$ to $\mathcal{W}$ is written $\mathcal{L}(\mathcal{V}, \mathcal{W})$ or $\hom_𝖪 (\mathcal{V}, \mathcal{W})$ where $𝖪$ is the ground field (or division ring for generalization to modules).

Sum of maps

Let $\Phi_i ∈ \mathcal{L}(\mathcal{V}, \mathcal{W})$ be over $𝖥$. The sum of linear maps is defined:

Where $\vec{v} ∈ \mathcal{V}$.

Scalar multiplication of maps

Let $σ_i ∈ 𝖥$. The scalar multiplication of linear maps is defined:

Discussion

The elements of $\mathcal{L}(\mathcal{V}, \mathcal{W})$ form a vector space under the sum and scaling of linear maps.

Dimension of $\mathcal{L}(\mathcal{V}, \mathcal{W})$

If $\Phi : \mathcal{V} → \mathcal{W}$ is a linear map between finite-dimensional vector spaces, then

Function Composition

Let $\Phi_i: \mathcal{V}_i → \mathcal{V}_{i+\!+}$ be linear.

Product of Linear Maps

Let $\Phi_i : \mathcal{V} → \mathcal{W}$, and $Ψ_i : \mathcal{W} → \mathcal{X}$, and $Ω_i : \mathcal{X} → \mathcal{Y}$ be linear. Then the product of linear maps is defined as the composition of compatible functions.

Where $\vec{v} ∈ \mathcal{V}$.

Ring of linear maps

Linear maps form a ring.

Matrix of a Linear Map

For some choice of basis in a vector space, a linear map may be represented as a matrix.

Identity Map

For any linear map $\Phi: \mathcal{V} → \mathcal{W}$ there also exists linear maps $\mathrm{I}_\mathcal{V}, \mathrm{I}_\mathcal{W}$ which act as the 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 $n$.

Discussion

$\mathrm{I}$ is also the identity map for its domain.

Invertible Maps

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

Discussion

• Linear isomorphism, invertibility, and bijectivity are all equivalent.
• An injective function is left-invertible.
• A surjective function is right-invertible.
• Matrices which are isomorphisms are also square.

If one interprets functions as having a direction from the source to the target, then right-cancellation in typical notation is an inversion prior to the map, and left-cancellation is an inversion after the map.

Morphisms on Vector Spaces

A morphism is a structure-preserving map between structured sets, such as vector spaces.

Homomorphism

A vector space homomorphism is an alternative definition of linear map. As a vector space is minimally equipped with a vector addition and scaling operation, those are the two operations which must be preserved by a vector space homomorphism.

Monomorphism (Injection)

A monomorphism is any left-cancellative homomorphism, and a vector space monomorphism is any injective linear map.

Equivalently, a linear injection $\Phi: \mathcal{V} ↪ \mathcal{W}$ can be defined as the condition of having a left-inverse $\Phi^{-1}_L: \mathcal{W} → \mathcal{V}$ such that:

In the Abelian context there is no difference between cancellation and the existence of an inverse, and thus a bimorphism is an isomorphism. The rest of the article won’t distinguish between the two concepts.

Epimorphism (Surjection)

An epimorphism is any right-cancellative homomorphism, and a vector space epimorphism is a linear surjection. Equivalently, a linear surjection $\Phi: \mathcal{V} ↠ \mathcal{W}$ can be defined as the condition of having a right-inverse $\Phi^{-1}: \mathcal{U} → \mathcal{V}$ such that:

Isomorphism (Bijection)

A linear isomorphism is any bijective linear map.

Endomorphism

A linear endomorphism is any linear map from a vector space to itself, and the set of all endomorphisms may be denoted $\mathcal{L}(\mathcal{V})$. In the context of matrices, a square matrix is equivalent to an endomorphism.

Automorphism

A vector space automorphism is any linear self-bijection, and in the context of matrices an invertible square matrix is equivalent to an automorphism. For matrices all isomorphisms are automorphisms, but in a more abstract context $\mathsf{R}^2 → \mathsf{C}$ is not an endomorphism.

graph TD
	 auto[auto];
	 endo[endo];
	 iso[iso];
	 homo[homo];
	 epi[epi];
	 mono[mono];
	 
   auto --> endo
   auto --> iso
      endo --> homo
      iso --> mono
      iso --> epi
         mono --> homo
         epi --> homo

The set of all automorphisms on an $n$-dimensional vector space $\mathcal{V}$ over a field $𝖥$ is known as the general linear group, and is denoted $\operatorname{GL}(n, 𝖥)$, or more generally $\operatorname{GL}(\mathcal{V})$.

In other words, the general linear group is another name for the automorphism group on $\mathcal{V}$.

Discussion

• If $\dim \mathcal{V} < \dim \mathcal{W}$ then $\Phi$ cannot be surjective.
• If $\dim \mathcal{V} > \dim \mathcal{W}$ then $\Phi$ cannot be injective.
• If $\dim \mathcal{V} ≠ \dim \mathcal{W}$ then $\Phi$ cannot be bijective.
• $\Phi$ is surjective iff $\dim \mathcal{V} = \operatorname{rank} \Phi$.
• $\Phi$ is injective iff $\operatorname{nullity} \Phi = 0$.
• For endomorphisms over finite-dimensional vector spaces $𝖥^n → 𝖥^n$, surjectivity, injectivity, and bijectivity are all equivalent. This statement can also be applied to finitely-generated modules.

Linear Forms

Let $\mathcal{V}$ be a vector space over $𝖥$.

• A linear form or functional is any linear map $\mathcal{V} → 𝖥$.
• The algebraic dual space (denoted $\mathcal{V}^*$) is the set of all linear forms from $\mathcal{V}$.
• A function $\mathcal{V} × \mathcal{V}^* → 𝖥$ is known as a natural pairing.

Discussion

• A linear form assigns every vector in $\mathcal{V}$ to a scalar in $𝖥$.
• A linear form either maps to $0$ or is surjective.
• In the finite-dimensional case $\mathcal{V}$ and $\mathcal{V}^*$ are bijective, but in the infinite case this is never true.

Transposition

Let $\Phi: \mathcal{V} → \mathcal{W}$ be a linear map. Then there is a dual map $\Phi^*: \mathcal{V}^* → \mathcal{W}^*$ such that:

Restriction of Linear Map to Subspace

Let $\Phi: \mathcal{V} → \mathcal{W}$ 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 $\mathcal{V}_1$.