Everything about Linear Operators totally explained

In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The term "linear transformation" is in particularly common use, especially for linear maps from a vector space to itself (endomorphisms). In the language of abstract algebra, a linear map is a homomorphism of vector spaces, or a morphism in the category of vector spaces over a given field.

Definition and first consequences

Let V and W be vector spaces over the same field K. A function f : V → W is said to be a linear map if for any two vectors x and y in V and any scalar a in K, the following two conditions are satisfied:

$f(x+y)=f(x)+f(y) ,$	additivity
$f(ax)=af(x) ,$	homogeneity

This is equivalent to requiring that for any vectors x₁, ..., x_m and scalars a₁, ..., a_m, the equality ť

f(a_1 x_1+cdots+a_m x_m)=a_1 f(x_1)+cdots+a_m f(x_m)

holds.
   Occasionally, V and W can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of "linear". If V and W are considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map C → C, but it isn't C-linear.
   A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional.
   It immediately follows from the definition that f(0) = 0. Hence linear maps are sometimes called homogeneous linear maps (see linear function).

Examples

The identity map and zero map are linear.

For real numbers, the map

xmapsto x^2

isn't linear.

For real numbers, the map

xmapsto x+1

isn't linear.

If A is an m × n matrix, then A defines a linear map from Rⁿ to R^m by sending the column vector x ∈ Rⁿ to the column vector Ax ∈ R^m. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the following section.

The integral yields a linear map from the space of all real-valued integrable functions on some interval to R

Differentiation is a linear map from the space of all differentiable functions to the space of all functions.

If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f : V → W to dim_F(W)-by-dim_F(V) matrices in the way described in the sequel are themselves linear maps.

Matrices

If V and W are finite-dimensional, and one has chosen bases in those spaces, then every linear map from V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear maps: if A is a real m-by-n matrix, then the rule f(x) = Ax describes a linear map Rⁿ → R^m (see Euclidean space).
Let

(f ))

dim(V ) , The number dim(im(f)) is also called the rank of f and written as rk(f), or sometimes, ρ(f); the number dim(ker(f)) is called the nullity of f and written as ν(f). If V and W are finite dimensional, bases have been chosen and f is represented by the matrix A, then the rank and nullity of f are equal to the rank and nullity of the matrix A, respectively.

Algebraic classifications of linear transformations

No classification of linear maps could hope to be exhaustive. The following incomplete list enumerates some important classifications that don't require any additional structure on the vector space.
Let V and W denote vector spaces over a field, F. Let T:V → W be a linear map.

T is said to be injective or a monomorphism if any of the following equivalent conditions are true:

T is one-to-one as a map of sets.
ker T 0
T is monic or left-cancellable, which is to say, for any vector space U and any pair of linear maps R:U → V and S:U → V, the equation TR=TS implies R=S.
T is left-invertible, which is to say there exists a linear map S:W → V such that ST is the identity map on V.

T is said to be surjective or an epimorphism if any of the following equivalent conditions are true:

T is onto as a map of sets.
coker T = 0
T is epic or right-cancellable, which is to say, for any vector space U and any pair of linear maps R:W → U and S:W → U, the equation RT=ST implies R=S.
T is right-invertible, which is to say there exists a linear map S:W → V such that TS is the identity map on V.

T is said to be an isomorphism if it's both left- and right-invertible. This is equivalent to T being both one-to-one and onto (a bijection of sets) or also to T being both epic and monic, and so being a bimorphism.

If T: V → V is an endomorphism, then:

If, for some positive integer n, the n-th iterate of T, Tⁿ, is identically zero, then T is said to be nilpotent.
If T T = T, then T is said to be idempotent
If T = k I, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map.

Continuity

A linear operator between topological vector spaces, for example normed spaces, may also be continuous and therefore be a continuous linear operator. On a normed space, a linear operator is continuous if and only if it's bounded, for example, when the domain is finite-dimensional. If the domain is infinite-dimensional, then there may be discontinuous linear operators. An example of an unbounded, hence not continuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values).

Applications

A specific application of linear maps is in the field of computational neuroscience. An example of a system being modeled is the innervation of V1 (primary visual cortex) by the retina. This transformation is called the logmap transformation. This kind of transformation is known as a domain coordinate transformation and provides a mathematical model of how neural states can be conferred within the system (CNS and PNS), when a change of state is required, such as from the retina to V1 as previously mentioned.
Another specific application is for geometric transformations, such as those performed in computer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix.
Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.

Further Information

Get more info on 'Linear Operators'.

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