Linear map

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 expression "linear transformation" is in particularly common use, especially for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides with the definition of linear map.

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.

1 Definition and first consequences
2 Examples
3 Matrices
4 Examples of linear transformation matrices
5 Forming new linear maps from given ones
6 Endomorphisms and automorphisms
7 Kernel, image and the rank-nullity theorem
8 Algebraic classifications of linear transformations
9 Continuity
10 Applications
11 See also
12 References

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 of degree 1

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

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

holds.

It immediately follows from the definition that f(0) = 0.

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 is not C-linear.

A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional.

Examples

The identity map and zero map are linear.

For real numbers, the map $x\mapsto x^2$ is not linear.

For real numbers, the map $x\mapsto x+1$ is not 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 $\{v_1, \cdots, v_n\}$ be a basis for V. Then every vector v in V is uniquely determined by the coefficients $c_1, \cdots, c_n$ in

$c_1 v_1+\cdots+c_n v_n.$

If f : V → W is a linear map,

$f(c_1 v_1+\cdots+c_n v_n)=c_1 f(v_1)+\cdots+c_n f(v_n),$

which implies that the function f is entirely determined by the values of $f(v_1),\cdots,f(v_n).$

Now let $\{w_1, \dots, w_m\}$ be a basis for W. Then we can represent the values of each $f(v_j)$ as

$f(v_j)=a_{1j} w_1 + \cdots + a_{mj} w_m.$

Thus, the function f is entirely determined by the values of $a_{i,j}.$

If we put these values into an m-by-n matrix M, then we can conveniently use it to compute the value of f for any vector in V. For if we place the values of $c_1, \cdots, c_n$ in an n-by-1 matrix C, we have MC = f(v).

A single linear map may be represented by many matrices. This is because the values of the elements of the matrix depend on the bases that are chosen.

Examples of linear transformation matrices

Some special cases of linear transformations of two-dimensional space R² are illuminating:

rotation by 90 degrees counterclockwise:

$A=\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$
rotation by θ degrees counterclockwise:

$A=\begin{bmatrix}\cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)\end{bmatrix}$
reflection against the x axis:

$A=\begin{bmatrix}1 & 0\\ 0 & -1\end{bmatrix}$
scaling by 2 in all directions:

$A=\begin{bmatrix}2 & 0\\ 0 & 2\end{bmatrix}$
vertical shear mapping:

$A=\begin{bmatrix}1 & m\\ 0 & 1\end{bmatrix}$
squeezing:

$A=\begin{bmatrix}k & 0\\ 0 & 1/k\end{bmatrix}$
projection onto the y axis:

$A=\begin{bmatrix}0 & 0\\ 0 & 1\end{bmatrix}$

Forming new linear maps from given ones

The composition of linear maps is linear: if f : V → W and g : W → Z are linear, then so is g o f : V → Z.

The inverse of a linear map, when defined, is again a linear map.

If f₁ : V → W and f₂ : V → W are linear, then so is their sum f₁ + f₂ (which is defined by (f₁ + f₂)(x) = f₁(x) + f₂(x)).

If f : V → W is linear and a is an element of the ground field K, then the map af, defined by (af)(x) = a (f(x)), is also linear.

Thus the set L(V,W) of linear maps from V to W itself forms a vector space over K, sometimes denoted Hom(V,W). Furthermore, in the case that V=W, this vector space (denoted End(V)) is an associative algebra under composition of maps, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.

Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication, the addition of linear maps corresponds to the matrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

Endomorphisms and automorphisms

A linear transformation f : V → V is an endomorphism of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is the identity map id : V → V.

An endomorphism of V that is also an isomorphism is called an automorphism of V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a group, the automorphism group of V which is denoted by Aut(V) or GL(V). Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V).

If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n by n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n by n invertible matrices with entries in K.

Kernel, image and the rank-nullity theorem

If f : V → W is linear, we define the kernel and the image or range of f by

$\operatorname{\ker}(f)=\{\,x\in V:f(x)=0\,\}$

$\operatorname{im}(f)=\{\,w\in W:w=f(x),x\in V\,\}$

ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula, known as the rank-nullity theorem, is often useful:

$\dim(\ker( f )) + \dim(\operatorname{im}( f )) = \dim( V ) \,$

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 do not 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 is 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

Main article: Discontinuous linear map

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 is 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, while the derivative of 0 is 0).

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.

References

Halmos, Paul R. (1974), Finite-dimensional vector spaces, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90093-3

Topics related to linear algebra

Scalar · Vector · Vector space · Vector projection · Linear span · Linear map · Linear projection · Linear independence · Linear combination · Basis · Column space · Row space · Dual space · Orthogonality · Rank · Minor · Kernel (matrix) · Eigenvalue, eigenvector and eigenspace · Least squares regressions · Outer product · Inner product space · Cross product · Dot product · Transpose · Gram–Schmidt process · Matrix decomposition