Linear map explained

V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.

If a linear map is a bijection then it is called a . In the case where

V = W , a linear map is called a linear endomorphism. Sometimes the term refers to this case,^[1] but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W ), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis.^[2] Sometimes the term linear function has the same meaning as linear map, while in analysis it does not.

A linear map from

V to W always maps the origin of V to the origin of W . Moreover, it maps linear subspaces in V onto linear subspaces in W (possibly of a lower dimension);^[3] for example, it maps a plane through the origin in V to either a plane through the origin in W , a line through the origin in W , or just the origin in W . Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

In the language of category theory, linear maps are the morphisms of vector spaces, and they form a category equivalent to the one of matrices.

Definition and first consequences

Let

V and W be vector spaces over the same field K . A function f: V \to W is said to be a linear map if for any two vectors $\backslash mathbf,\; \backslash mathbf\; \backslash in\; V$ and any scalar c \in K the following two conditions are satisfied:

• Additivity / operation of addition $$f(\backslash mathbf\; +\; \backslash mathbf)\; =\; f(\backslash mathbf)\; +\; f(\backslash mathbf)$$
• Homogeneity of degree 1 / operation of scalar multiplication $$f(c\; \backslash mathbf)\; =\; c\; f(\backslash mathbf)$$

Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.

By the associativity of the addition operation denoted as +, for any vectors $\backslash mathbf\_1,\; \backslash ldots,\; \backslash mathbf\_n\; \backslash in\; V$ and scalars $c\_1,\; \backslash ldots,\; c\_n\; \backslash in\; K,$ the following equality holds:^{[4] [5]} $$f(c\_1\; \backslash mathbf\_1\; +\; \backslash cdots\; +\; c\_n\; \backslash mathbf\_n)\; =\; c\_1\; f(\backslash mathbf\_1)\; +\; \backslash cdots\; +\; c\_n\; f(\backslash mathbf\_n).$$ Thus a linear map is one which preserves linear combinations.

Denoting the zero elements of the vector spaces

V and W by $\backslash mathbf\_V$ and $\backslash mathbf\_W$ respectively, it follows that $f(\backslash mathbf\_V)\; =\; \backslash mathbf\_W.$ Let c = 0 and $\backslash mathbf\; \backslash in\; V$ in the equation for homogeneity of degree 1:$$f(\backslash mathbf\_V)\; =\; f(0\backslash mathbf)\; =\; 0f(\backslash mathbf)\; =\; \backslash mathbf\_W.$$

A linear map

V \to K with K viewed as a one-dimensional vector space over itself is called a linear functional.^[6]

These statements generalize to any left-module $\_R\; M$ over a ring

R without modification, and to any right-module upon reversing of the scalar multiplication.

Examples

• A prototypical example that gives linear maps their name is a function

f: R \to R : x \mapsto cx , of which the graph is a line through the origin.^[7]

• More generally, any homothety $\backslash mathbf\; \backslash mapsto\; c\backslash mathbf$ centered in the origin of a vector space is a linear map (here is a scalar).
• The zero map $\backslash mathbf\; x\; \backslash mapsto\; \backslash mathbf\; 0$ between two vector spaces (over the same field) is linear.
• The identity map on any module is a linear operator.
• For real numbers, the map $x\; \backslash mapsto\; x^2$ is not linear.
• For real numbers, the map $x\; \backslash mapsto\; x\; +\; 1$ is not linear (but is an affine transformation).
• If

A is a m x n real matrix, then A defines a linear map from \R ^{n} to \R ^{m} by sending a column vector x \in \R ^{n} to the column vector A x \in \R ^{m} . Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the, below.

• If $f:\; V\; \backslash to\; W$ is an isometry between real normed spaces such that $f(0)\; =\; 0$ then

f is a linear map. This result is not necessarily true for complex normed space.

• Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a linear operator on the space of all smooth functions (a linear operator is a linear endomorphism, that is, a linear map with the same domain and codomain). Indeed, $$\backslash frac\; \backslash left(a\; f(x)\; +\; b\; g(x)\; \backslash right)\; =\; a\; \backslash frac\; +\; b\; \backslash frac.$$
• A definite integral over some interval is a linear map from the space of all real-valued integrable functions on to

\R . Indeed, $$\backslash int\_u^v\; \backslash left(af(x)\; +\; bg(x)\backslash right)\; dx\; =\; a\backslash int\_u^v\; f(x)\; dx\; +\; b\backslash int\_u^v\; g(x)\; dx\; .$$

• An indefinite integral (or antiderivative) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on

\R to the space of all real-valued, differentiable functions on \R . Without a fixed starting point, the antiderivative maps to the quotient space of the differentiable functions by the linear space of constant functions.

• If

V and W are finite-dimensional vector spaces over a field, of respective dimensions and, then the function that maps linear maps $f:\; V\; \backslash to\; W$ to matrices in the way described in (below) is a linear map, and even a linear isomorphism.

• The expected value of a random variable (which is in fact a function, and as such an element of a vector space) is linear, as for random variables

X and Y we have E[X + Y] = E[X] + E[Y] and E[aX] = aE[X] , but the variance of a random variable is not linear.

Linear extensions

Often, a linear map is constructed by defining it on a subset of a vector space and then to the linear span of the domain. Suppose

X and Y are vector spaces and f : S \to Y is a function defined on some subset S \subseteq X. Then a of f to X, if it exists, is a linear map F : X \to Y defined on X that extends f ^[8] (meaning that F(s) = f(s) for all s \in S ) and takes its values from the codomain of f. When the subset S is a vector subspace of X then a (Y -valued) linear extension of f to all of X is guaranteed to exist if (and only if) f : S \to Y is a linear map. In particular, if f has a linear extension to \operatorname{span} S, then it has a linear extension to all of X.

The map

f : S \to Y can be extended to a linear map F : \operatorname{span} S \to Y if and only if whenever n > 0 is an integer, c _{1,} \ldots, c _{n} are scalars, and s _{1,} \ldots, s _{n} \in S are vectors such that 0 = c _{1} s _{1} + … + c _{n} s _{n,} then necessarily 0 = c _{1} f\left(s _{1\right)} + … + c _{n} f\left(s _{n\right).} If a linear extension of f : S \to Y exists then the linear extension F : \operatorname{span} S \to Y is unique and$$F\backslash left(c\_1\; s\_1\; +\; \backslash cdots\; c\_n\; s\_n\backslash right)\; =\; c\_1\; f\backslash left(s\_1\backslash right)\; +\; \backslash cdots\; +\; c\_n\; f\backslash left(s\_n\backslash right)$$holds for all n, c _{1,} \ldots, c _{n,} and s _{1,} \ldots, s _{n} as above. If S is linearly independent then every function f : S \to Y into any vector space has a linear extension to a (linear) map \operatorname{span} S \to Y (the converse is also true).

For example, if

X = \R ^{2} and Y = \R then the assignment ( 1 , 0 ) \to -1 and ( 0 , 1 ) \to 2 can be linearly extended from the linearly independent set of vectors S : = \{( 1 , 0 ), ( 0 , 1 )\} to a linear map on \operatorname{span}\{( 1 , 0 ), ( 0 , 1 )\} = \R ^{2.} The unique linear extension F : \R ^{2} \to \R is the map that sends (x, y) = x ( 1 , 0 ) + y ( 0 , 1 ) \in \R ^{2} to$$F(x,\; y)\; =\; x\; (-1)\; +\; y\; (2)\; =\; -\; x\; +\; 2\; y.$$f defined on a vector subspace of a real or complex vector space X has a linear extension to all of X. Indeed, the Hahn–Banach dominated extension theorem even guarantees that when this linear functional f is dominated by some given seminorm p : X \to \R (meaning that |f(m)| \leq p(m) holds for all m in the domain of f ) then there exists a linear extension to X that is also dominated by p.

Matrices

See main article: Transformation matrix.

If

V and W are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from V to W can be represented by a matrix.^[9] This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if A is a real m x n matrix, then f( x) = A x describes a linear map \R ^{n} \to \R ^{m} (see Euclidean space).

Let

\{ v _{1,} \ldots , v _{n} \} be a basis for V . Then every vector v \in V is uniquely determined by the coefficients c _{1,} \ldots , c _{n} in the field \R :$$\backslash mathbf\; =\; c\_1\; \backslash mathbf\_1\; +\; \backslash cdots\; +\; c\_n\; \backslash mathbf\; \_n.$$

If $f:\; V\; \backslash to\; W$ is a linear map,$$f(\backslash mathbf)\; =\; f(c\_1\; \backslash mathbf\_1\; +\; \backslash cdots\; +\; c\_n\; \backslash mathbf\_n)\; =\; c\_1\; f(\backslash mathbf\_1)\; +\; \backslash cdots\; +\; c\_n\; f\backslash left(\backslash mathbf\_n\backslash right),$$

which implies that the function f is entirely determined by the vectors

f( v _{1),} \ldots , f( v _{n)} . Now let \{ w _{1,} \ldots , w _{m} \} be a basis for W . Then we can represent each vector f( v _{j)} as$$f\backslash left(\backslash mathbf\_j\backslash right)\; =\; a\_\; \backslash mathbf\_1\; +\; \backslash cdots\; +\; a\_\; \backslash mathbf\_m.$$

Thus, the function

f is entirely determined by the values of a _{ij} . If we put these values into an m x n matrix M , then we can conveniently use it to compute the vector output of f for any vector in V . To get M , every column j of M is a vector$$\backslash begin\; a\_\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; a\_\; \backslash end$$corresponding to f( v _{j)} as defined above. To define it more clearly, for some column j that corresponds to the mapping f( v _{j)} ,where M is the matrix of f . In other words, every column j = 1 , \ldots, n has a corresponding vector f( v _{j)} whose coordinates a _{1j} , … , a _{mj} are the elements of column j . A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen.

The matrices of a linear transformation can be represented visually:

1. Matrix for $T$ relative to $B$: $A$
2. Matrix for $T$ relative to $B\text{'}$: $A\text{'}$
3. Transition matrix from $B\text{'}$ to $B$: $P$
4. Transition matrix from $B$ to $B\text{'}$: $P^$

Such that starting in the bottom left corner $\backslash left[\backslash mathbf\{v\}\backslash right]\_$ and looking for the bottom right corner $\backslash left[T\backslash left(\backslash mathbf\{v\}\backslash right)\backslash right]\_$, one would left-multiply—that is, $A\text{'}\backslash left[\backslash mathbf\{v\}\backslash right]\_\; =\; \backslash left[T\backslash left(\backslash mathbf\{v\}\backslash right)\backslash right]\_$. The equivalent method would be the "longer" method going clockwise from the same point such that $\backslash left[\backslash mathbf\{v\}\backslash right]\_$ is left-multiplied with $P^AP$, or $P^AP\backslash left[\backslash mathbf\{v\}\backslash right]\_\; =\; \backslash left[T\backslash left(\backslash mathbf\{v\}\backslash right)\backslash right]\_$.

Examples in two dimensions

In two-dimensional space R² linear maps are described by 2 × 2 matrices. These are some examples:

• rotation
  • by 90 degrees counterclockwise:
  • by an angle θ counterclockwise:
• reflection
  • through the x axis:
  • through the y axis:
  • through a line making an angle θ with the origin:
• scaling by 2 in all directions:
• horizontal shear mapping:
• skew of the y axis by an angle θ:
• squeeze mapping:
• projection onto the y axis:

If a linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a conformal linear transformation.

Vector space of linear maps

The composition of linear maps is linear: if

f: V \to W and $g:\; W\; \backslash to\; Z$ are linear, then so is their composition $g\; \backslash circ\; f:\; V\; \backslash to\; Z$. It follows from this that the class of all vector spaces over a given field K, together with K-linear maps as morphisms, forms a category.

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

If $f\_1:\; V\; \backslash to\; W$ and $f\_2:\; V\; \backslash to\; W$ are linear, then so is their pointwise sum

f _{1} + f _{2} , which is defined by (f _{1} + f _{2)(x)=} f _{1(x)+} f _{2(x)} .

If $f:\; V\; \backslash to\; W$ is linear and $\backslash alpha$ is an element of the ground field $K$, then the map $\backslash alpha\; f$, defined by $(\backslash alpha\; f)(\backslash mathbf\; x)\; =\; \backslash alpha\; (f(\backslash mathbf\; x))$, is also linear.

Thus the set $\backslash mathcal(V,\; W)$ of linear maps from $V$ to $W$ itself forms a vector space over $K$,^[10] sometimes denoted $\backslash operatorname(V,\; W)$.^[11] Furthermore, in the case that $V\; =\; W$, this vector space, denoted $\backslash operatorname(V)$, 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

See main article: Endomorphism and Automorphism. A linear transformation $f\; :\; V\; \backslash to\; V$ is an endomorphism of $V$; the set of all such endomorphisms $\backslash operatorname(V)$ 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 $\backslash operatorname:\; V\; \backslash to\; V$.

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 $\backslash operatorname(V)$ or $\backslash operatorname(V)$. Since the automorphisms are precisely those endomorphisms which possess inverses under composition, $\backslash operatorname(V)$ is the group of units in the ring $\backslash operatorname(V)$.

If $V$ has finite dimension $n$, then $\backslash operatorname(V)$ is isomorphic to the associative algebra of all $n\; \backslash times\; n$ matrices with entries in $K$. The automorphism group of $V$ is isomorphic to the general linear group $\backslash operatorname(n,\; K)$ of all $n\; \backslash times\; n$ invertible matrices with entries in $K$.

Kernel, image and the rank–nullity theorem

See main article: Kernel (linear algebra), Image (mathematics) and Rank of a matrix. If $f:\; V\; \backslash to\; W$ is linear, we define the kernel and the image or range of $f$ by

$\backslash ker(f)$ is a subspace of $V$ and $\backslash operatorname(f)$ is a subspace of $W$. The following dimension formula is known as the rank–nullity theorem:$$\backslash dim(\backslash ker(f))\; +\; \backslash dim(\backslash operatorname(f))\; =\; \backslash dim(V).$$

The number $\backslash dim(\backslash operatorname(f))$ is also called the rank of $f$ and written as $\backslash operatorname(f)$, or sometimes, $\backslash rho(f)$;^{[12] [13]} the number $\backslash dim(\backslash ker(f))$ is called the nullity of $f$ and written as $\backslash operatorname(f)$ or $\backslash nu(f)$. 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.

Cokernel

See main article: Cokernel.

A subtler invariant of a linear transformation $f:\; V\; \backslash to\; W$ is the cokernel, which is defined as$$\backslash operatorname(f)\; :=\; W/f(V)\; =\; W/\backslash operatorname(f).$$

This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the co-kernel is a quotient space of the target. Formally, one has the exact sequence$$0\; \backslash to\; \backslash ker(f)\; \backslash to\; V\; \backslash to\; W\; \backslash to\; \backslash operatorname(f)\; \backslash to\; 0.$$

These can be interpreted thus: given a linear equation f(v) = w to solve,

• the kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the number of degrees of freedom in the space of solutions, if it is not empty;
• the co-kernel is the space of constraints that the solutions must satisfy, and its dimension is the maximal number of independent constraints.

The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space W/f(V) is the dimension of the target space minus the dimension of the image.

As a simple example, consider the map f: R² → R², given by f(x, y) = (0, y). Then for an equation f(x, y) = (a, b) to have a solution, we must have a = 0 (one constraint), and in that case the solution space is (x, b) or equivalently stated, (0, b) + (x, 0), (one degree of freedom). The kernel may be expressed as the subspace (x, 0) < V: the value of x is the freedom in a solution – while the cokernel may be expressed via the map W → R, $(a,\; b)\; \backslash mapsto\; (a)$: given a vector (a, b), the value of a is the obstruction to there being a solution.

An example illustrating the infinite-dimensional case is afforded by the map f: R^∞ → R^∞, $\backslash left\backslash \; \backslash mapsto\; \backslash left\backslash$ with b₁ = 0 and b_{n + 1} = a_n for n > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel ($\backslash aleph\_0\; +\; 0\; =\; \backslash aleph\_0\; +\; 1$), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an endomorphism have the same dimension (0 ≠ 1). The reverse situation obtains for the map h: R^∞ → R^∞, $\backslash left\backslash \; \backslash mapsto\; \backslash left\backslash$ with c_n = a_{n + 1}. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.

Index

For a linear operator with finite-dimensional kernel and co-kernel, one may define index as:$$\backslash operatorname(f)\; :=\; \backslash dim(\backslash ker(f))\; -\; \backslash dim(\backslash operatorname(f)),$$namely the degrees of freedom minus the number of constraints.

For a transformation between finite-dimensional vector spaces, this is just the difference dim(V) − dim(W), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom.

The index of an operator is precisely the Euler characteristic of the 2-term complex 0 → V → W → 0. In operator theory, the index of Fredholm operators is an object of study, with a major result being the Atiyah–Singer index theorem.^[14]

Algebraic classifications of linear transformations

No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.

Let and denote vector spaces over a field and let be a linear map.

Monomorphism

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

1. is one-to-one as a map of sets.
   1. 1. is monic or left-cancellable, which is to say, for any vector space and any pair of linear maps and, the equation implies .
2. is left-invertible, which is to say there exists a linear map such that is the identity map on .

Epimorphism

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

1. is onto as a map of sets.
   1. is epic or right-cancellable, which is to say, for any vector space and any pair of linear maps and, the equation implies .
2. is right-invertible, which is to say there exists a linear map such that is the identity map on .

Isomorphism

is said to be an isomorphism if it is both left- and right-invertible. This is equivalent to being both one-to-one and onto (a bijection of sets) or also to being both epic and monic, and so being a bimorphism.If is an endomorphism, then:

• If, for some positive integer, the -th iterate of,, is identically zero, then is said to be nilpotent.
• If, then is said to be idempotent
• If, where is some scalar, then is said to be a scaling transformation or scalar multiplication map; see scalar matrix.

Change of basis

See main article: Basis (linear algebra) and Change of basis. Given a linear map which is an endomorphism whose matrix is A, in the basis B of the space it transforms vector coordinates [u] as [v] = A[u]. As vectors change with the inverse of B (vectors coordinates are contravariant) its inverse transformation is [v] = B[v'].

Substituting this in the first expression$$B\backslash left[v\text{'}\backslash right]\; =\; AB\backslash left[u\text{'}\backslash right]$$hence$$\backslash left[v\text{'}\backslash right]\; =\; B^AB\backslash left[u\text{'}\backslash right]\; =\; A\text{'}\backslash left[u\text{'}\backslash right].$$

Therefore, the matrix in the new basis is A′ = B⁻¹AB, being B the matrix of the given basis.

Therefore, linear maps are said to be 1-co- 1-contra-variant objects, or type (1, 1) tensors.

Continuity

See main article: Continuous linear operator and Discontinuous linear map.

A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional.^[15] An infinite-dimensional domain may have discontinuous linear operators.

An example of an unbounded, hence discontinuous, 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). For a specific example, converges to 0, but its derivative does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).

Applications

A specific application of linear maps 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. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames.

Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.

See also

• • • • • • • • • Category of matrices
• Quasilinearization

Bibliography

• Book: Axler, Sheldon Jay. Linear Algebra Done Right. Springer. 2015. 978-3-319-11079-0. 3rd. Sheldon Axler.
• Book: Bronshtein . I. N. . Semendyayev . K. A. . 2004 . . 4th . New York . Springer-Verlag . 3-540-43491-7.
• Book: Halmos, Paul Richard. Finite-Dimensional Vector Spaces. Springer. 1974. 0-387-90093-4. 2nd. Paul Halmos. 1958.
• Book: Horn . Roger A. . Johnson . Charles R. . Matrix Analysis . Second . . 978-0-521-83940-2 . 2013 .
• Book: Katznelson, Yitzhak. A (Terse) Introduction to Linear Algebra. Katznelson. Yonatan R.. American Mathematical Society. 2008. 978-0-8218-4419-9. Yitzhak Katznelson.
• Book: Kubrusly, Carlos. Elements of operator theory. Birkhäuser. Boston. 2001. 978-1-4757-3328-0. 754555941.
  • • Book: Rudin, Walter. Walter Rudin. 1976. 3rd. Walter Rudin Student Series in Advanced Mathematics. New York. Principles of Mathematical Analysis. McGraw–Hill. 978-0-07-054235-8.
      • • Book: Tu, Loring W.. An Introduction to Manifolds. Springer. 2011. 978-0-8218-4419-9. 2nd. Loring W. Tu.

Notes and References

1. "Linear transformations of into are often called linear operators on ."
2. Let and be two real vector spaces. A mapping a from into Is called a 'linear mapping' or 'linear transformation' or 'linear operator' [...] from into, if
   $a(\backslash mathbf\; u\; +\; \backslash mathbf\; v)\; =\; a\; \backslash mathbf\; u\; +\; a\; \backslash mathbf\; v$ for all $\backslash mathbf\; u,\backslash mathbf\; v\; \backslash in\; V$,
   $a(\backslash lambda\; \backslash mathbf\; u)\; =\; \backslash lambda\; a\; \backslash mathbf\; u$ for all u \in V and all real .
3. 
   Here are some properties of linear mappings $\backslash Lambda:\; X\; \backslash to\; Y$ whose proofs are so easy that we omit them; it is assumed that $A\; \backslash subset\; X$ and $B\; \backslash subset\; Y$:
4. . Suppose now that and are vector spaces over the same scalar field. A mapping $\backslash Lambda:\; X\; \backslash to\; Y$ is said to be linear if $\backslash Lambda(\backslash alpha\; \backslash mathbf\; x\; +\; \backslash beta\; \backslash mathbf\; y)\; =\; \backslash alpha\; \backslash Lambda\; \backslash mathbf\; x\; +\; \backslash beta\; \backslash Lambda\; \backslash mathbf\; y$ for all $\backslash mathbf\; x,\; \backslash mathbf\; y\; \backslash in\; X$ and all scalars $\backslash alpha$ and $\backslash beta$. Note that one often writes $\backslash Lambda\; \backslash mathbf\; x$, rather than $\backslash Lambda(\backslash mathbf\; x)$, when $\backslash Lambda$ is linear.
5. . A mapping of a vector space into a vector space is said to be a linear transformation if: $A\backslash left(\backslash mathbf\_1\; +\; \backslash mathbf\_2\backslash right)\; =\; A\backslash mathbf\_1\; +\; A\backslash mathbf\_2,\backslash \; A(c\backslash mathbf)\; =\; c\; A\backslash mathbf$ for all $\backslash mathbf,\; \backslash mathbf\_1,\; \backslash mathbf\_2\; \backslash in\; X$ and all scalars . Note that one often writes $A\backslash mathbf$ instead of $A(\backslash mathbf)$ if is linear.
6. . Linear mappings of onto its scalar field are called linear functionals.
7. Web site: terminology - What does 'linear' mean in Linear Algebra?. 2021-02-17. Mathematics Stack Exchange.
8. One map F is said to another map f if when f is defined at a point s, then so is F and F(s) = f(s).

9. Suppose $\backslash left\backslash$ and $\backslash left\backslash$ are bases of vector spaces and, respectively. Then every $A\; \backslash in\; L(X,\; Y)$ determines a set of numbers $a\_$ such that$$A\backslash mathbf\_j\; =\; \backslash sum\_^m\; a\_\backslash mathbf\_i\backslash quad\; (1\; \backslash leq\; j\; \backslash leq\; n).$$

   It is convenient to represent these numbers in a rectangular array of rows and columns, called an by matrix:

   Observe that the coordinates $a\_$ of the vector $A\backslash mathbf\_j$ (with respect to the basis $\backslash$) appear in the j^th column of $[A]$. The vectors $A\backslash mathbf\_j$ are therefore sometimes called the column vectors of $[A]$. With this terminology, the range of is spanned by the column vectors of $[A]$.

10. p. 52, § 3.3
11. , p. 19, § 3.1
12. p. 52, § 2.5.1
13. p. 90, § 50

14. "The main question in index theory is to provide index formulas for classes of Fredholm operators ... Index theory has become a subject on its own only after M. F. Atiyah and I. Singer published their index theorems"

15. 1.18 Theorem Let $\backslash Lambda$ be a linear functional on a topological vector space . Assume $\backslash Lambda\; \backslash mathbf\; x\; \backslash neq\; 0$ for some $\backslash mathbf\; x\; \backslash in\; X$. Then each of the following four properties implies the other three: