
This article includes a list of general references, but it lacks sufficient corresponding inline citations.(December 2021) |
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping 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 linear isomorphism. In the case where , a linear map is called a linear endomorphism. Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that and are real vector spaces (not necessarily with ),[citation needed] or it can be used to emphasize that is a function space, which is a common convention in functional analysis. Sometimes the term linear function has the same meaning as linear map, while in analysis it does not.
A linear map from to always maps the origin of to the origin of . Moreover, it maps linear subspaces in onto linear subspaces in (possibly of a lower dimension); for example, it maps a plane through the origin in to either a plane through the origin in , a line through the origin in , or just the origin in . 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 and
be vector spaces over the same field
. A function
is said to be a linear map if for any two vectors
and any scalar
the following two conditions are satisfied:
- Additivity / operation of addition
- Homogeneity of degree 1 / operation of scalar multiplication
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 and scalars
the following equality holds:
Thus a linear map is one which preserves linear combinations.
Denoting the zero elements of the vector spaces and
by
and
respectively, it follows that
Let
and
in the equation for homogeneity of degree 1:
A linear map with
viewed as a one-dimensional vector space over itself is called a linear functional.
These statements generalize to any left-module over a ring
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
, of which the graph is a line through the origin.
- More generally, any homothety
centered in the origin of a vector space is a linear map (here c is a scalar).
- The zero map
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
is not linear.
- For real numbers, the map
is not linear (but is an affine transformation).
- If
is a
real matrix, then
defines a linear map from
to
by sending a column vector
to the column vector
. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the § Matrices, below.
- If
is an isometry between real normed spaces such that
then
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,
- A definite integral over some interval I is a linear map from the space of all real-valued integrable functions on I to
. Indeed,
- 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
to the space of all real-valued, differentiable functions on
. Without a fixed starting point, the antiderivative maps to the quotient space of the differentiable functions by the linear space of constant functions.
- If
and
are finite-dimensional vector spaces over a field F, of respective dimensions m and n, then the function that maps linear maps
to n × m matrices in the way described in § Matrices (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
and
we have
and
, but the variance of a random variable is not linear.
- The function
with
is a linear map. This function scales the
component of a vector by the factor
.
- The function
is additive: It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added:
- The function
is homogeneous: It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled:
Linear extensions
Often, a linear map is constructed by defining it on a subset of a vector space and then extending by linearity to the linear span of the domain. Suppose and
are vector spaces and
is a function defined on some subset
Then a linear extension of
to
if it exists, is a linear map
defined on
that extends
(meaning that
for all
) and takes its values from the codomain of
When the subset
is a vector subspace of
then a (
-valued) linear extension of
to all of
is guaranteed to exist if (and only if)
is a linear map. In particular, if
has a linear extension to
then it has a linear extension to all of
The map can be extended to a linear map
if and only if whenever
is an integer,
are scalars, and
are vectors such that
then necessarily
If a linear extension of
exists then the linear extension
is unique and
holds for all
and
as above. If
is linearly independent then every function
into any vector space has a linear extension to a (linear) map
(the converse is also true).
For example, if and
then the assignment
and
can be linearly extended from the linearly independent set of vectors
to a linear map on
The unique linear extension
is the map that sends
to
Every (scalar-valued) linear functional defined on a vector subspace of a real or complex vector space
has a linear extension to all of
Indeed, the Hahn–Banach dominated extension theorem even guarantees that when this linear functional
is dominated by some given seminorm
(meaning that
holds for all
in the domain of
) then there exists a linear extension to
that is also dominated by
Matrices
If and
are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from
to
can be represented by a matrix. This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if
is a real
matrix, then
describes a linear map
(see Euclidean space).
Let be a basis for
. Then every vector
is uniquely determined by the coefficients
in the field
:
If is a linear map,
which implies that the function f is entirely determined by the vectors . Now let
be a basis for
. Then we can represent each vector
as
Thus, the function is entirely determined by the values of
. If we put these values into an
matrix
, then we can conveniently use it to compute the vector output of
for any vector in
. To get
, every column
of
is a vector
corresponding to
as defined above. To define it more clearly, for some column
that corresponds to the mapping
,
where
is the matrix of
. In other words, every column
has a corresponding vector
whose coordinates
are the elements of column
. 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:
- Matrix for
relative to
:
- Matrix for
relative to
:
- Transition matrix from
to
:
- Transition matrix from
to
:
Such that starting in the bottom left corner and looking for the bottom right corner
, one would left-multiply—that is,
. The equivalent method would be the "longer" method going clockwise from the same point such that
is left-multiplied with
, or
.
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:
- by 90 degrees counterclockwise:
- reflection
- through the x axis:
- through the y axis:
- through a line making an angle θ with the origin:
- through the x axis:
- 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 and
are linear, then so is their composition
. 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 and
are linear, then so is their pointwise sum
, which is defined by
.
If is linear and
is an element of the ground field
, then the map
, defined by
, is also linear.
Thus the set of linear maps from
to
itself forms a vector space over
, sometimes denoted
. Furthermore, in the case that
, this vector space, denoted
, 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 is an endomorphism of
; the set of all such endomorphisms
together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field
(and in particular a ring). The multiplicative identity element of this algebra is the identity map
.
An endomorphism of that is also an isomorphism is called an automorphism of
. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of
forms a group, the automorphism group of
which is denoted by
or
. Since the automorphisms are precisely those endomorphisms which possess inverses under composition,
is the group of units in the ring
.
If has finite dimension
, then
is isomorphic to the associative algebra of all
matrices with entries in
. The automorphism group of
is isomorphic to the general linear group
of all
invertible matrices with entries in
.
Kernel, image and the rank–nullity theorem
If is linear, we define the kernel and the image or range of
by
is a subspace of
and
is a subspace of
. The following dimension formula is known as the rank–nullity theorem:
The number is also called the rank of
and written as
, or sometimes,
; the number
is called the nullity of
and written as
or
. If
and
are finite-dimensional, bases have been chosen and
is represented by the matrix
, then the rank and nullity of
are equal to the rank and nullity of the matrix
, respectively.
Cokernel
A subtler invariant of a linear transformation is the cokernel, which is defined as
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
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, : 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∞, with b1 = 0 and bn + 1 = an 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 (
), 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∞,
with cn = an + 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: 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.
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 V and W denote vector spaces over a field F and let T: V → W be a linear map.
Monomorphism
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 = {0V}
- dim(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.
Epimorphism
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 = {0W}
- 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 W.
Isomorphism
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 T2 = T, then T is said to be idempotent
- If T = kI, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map; see scalar matrix.
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 hence
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
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. 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, sin(nx)/n converges to 0, but its derivative cos(nx) 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
- Additive map – Z-module homomorphism
- Antilinear map – Conjugate homogeneous additive map
- Bent function – Special type of Boolean function
- Bounded operator – Linear transformation between topological vector spaces
- Cauchy's functional equation – Functional equation
- Continuous linear operator
- Linear functional – Linear map from a vector space to its field of scalars
- Linear isometry – Distance-preserving mathematical transformation
- Category of matrices
- Quasilinearization
Notes
- "Linear transformations of V into V are often called linear operators on V." Rudin 1976, p. 207
- Let V and W be two real vector spaces. A mapping a from V into W Is called a 'linear mapping' or 'linear transformation' or 'linear operator' [...] from V into W, if
for all
,
for all
and all real λ. Bronshtein & Semendyayev 2004, p. 316
- Rudin 1991, p. 14
Here are some properties of linear mappingswhose proofs are so easy that we omit them; it is assumed that
and
:
- If A is a subspace (or a convex set, or a balanced set) the same is true of
- If B is a subspace (or a convex set, or a balanced set) the same is true of
- In particular, the set:
is a subspace of X, called the null space of
.
- Rudin 1991, p. 14. Suppose now that X and Y are vector spaces over the same scalar field. A mapping
is said to be linear if
for all
and all scalars
and
. Note that one often writes
, rather than
, when
is linear.
- Rudin 1976, p. 206. A mapping A of a vector space X into a vector space Y is said to be a linear transformation if:
for all
and all scalars c. Note that one often writes
instead of
if A is linear.
- Rudin 1991, p. 14. Linear mappings of X onto its scalar field are called linear functionals.
- "terminology - What does 'linear' mean in Linear Algebra?". Mathematics Stack Exchange. Retrieved 2021-02-17.
- Wilansky 2013, pp. 21–26.
- Kubrusly 2001, p. 57.
- Schechter 1996, pp. 277–280.
- Rudin 1976, p. 210 Suppose
and
are bases of vector spaces X and Y, respectively. Then every
determines a set of numbers
such that
It is convenient to represent these numbers in a rectangular array of m rows and n columns, called an m by n matrix:
Observe that the coordinates
of the vector
(with respect to the basis
) appear in the jth column of
. The vectors
are therefore sometimes called the column vectors of
. With this terminology, the range of A is spanned by the column vectors of
.
- Axler (2015) p. 52, § 3.3
- Tu (2011), p. 19, § 3.1
- Horn & Johnson 2013, 0.2.3 Vector spaces associated with a matrix or linear transformation, p. 6
- Katznelson & Katznelson (2008) p. 52, § 2.5.1
- Halmos (1974) p. 90, § 50
- Nistor, Victor (2001) [1994], "Index theory", Encyclopedia of Mathematics, EMS Press: "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"
- Rudin 1991, p. 15 1.18 Theorem Let
be a linear functional on a topological vector space X. Assume
for some
. Then each of the following four properties implies the other three:
is continuous
- The null space
is closed.
is not dense in X.
is bounded in some neighbourhood V of 0.
- One map
is said to extend another map
if when
is defined at a point
then so is
and
Bibliography
- Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0.
- Bronshtein, I. N.; Semendyayev, K. A. (2004). Handbook of Mathematics (4th ed.). New York: Springer-Verlag. ISBN 3-540-43491-7.
- Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces (2nd ed.). Springer. ISBN 0-387-90093-4.
- Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (Second ed.). Cambridge University Press. ISBN 978-0-521-83940-2.
- Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9.
- Kubrusly, Carlos (2001). Elements of operator theory. Boston: Birkhäuser. ISBN 978-1-4757-3328-0. OCLC 754555941.
- Lang, Serge (1987), Linear Algebra (Third ed.), New York: Springer-Verlag, ISBN 0-387-96412-6
- Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259.
- Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). New York: McGraw–Hill. ISBN 978-0-07-054235-8.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
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This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations December 2021 Learn how and when to remove this message In mathematics and more specifically in linear algebra a linear map also called a linear mapping linear transformation vector space homomorphism or in some contexts linear function is a mapping V W displaystyle 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 linear isomorphism In the case where V W displaystyle V W a linear map is called a linear endomorphism Sometimes the term linear operator refers to this case but the term linear operator can have different meanings for different conventions for example it can be used to emphasize that V displaystyle V and W displaystyle W are real vector spaces not necessarily with V W displaystyle V W citation needed or it can be used to emphasize that V displaystyle V is a function space which is a common convention in functional analysis Sometimes the term linear function has the same meaning as linear map while in analysis it does not A linear map from V displaystyle V to W displaystyle W always maps the origin of V displaystyle V to the origin of W displaystyle W Moreover it maps linear subspaces in V displaystyle V onto linear subspaces in W displaystyle W possibly of a lower dimension for example it maps a plane through the origin in V displaystyle V to either a plane through the origin in W displaystyle W a line through the origin in W displaystyle W or just the origin in W displaystyle 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 consequencesLet V displaystyle V and W displaystyle W be vector spaces over the same field K displaystyle K A function f V W displaystyle f V to W is said to be a linear map if for any two vectors u v V textstyle mathbf u mathbf v in V and any scalar c K displaystyle c in K the following two conditions are satisfied Additivity operation of addition f u v f u f v displaystyle f mathbf u mathbf v f mathbf u f mathbf v Homogeneity of degree 1 operation of scalar multiplication f cu cf u displaystyle f c mathbf u cf mathbf u 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 u1 un V textstyle mathbf u 1 ldots mathbf u n in V and scalars c1 cn K textstyle c 1 ldots c n in K the following equality holds f c1u1 cnun c1f u1 cnf un displaystyle f c 1 mathbf u 1 cdots c n mathbf u n c 1 f mathbf u 1 cdots c n f mathbf u n Thus a linear map is one which preserves linear combinations Denoting the zero elements of the vector spaces V displaystyle V and W displaystyle W by 0V textstyle mathbf 0 V and 0W textstyle mathbf 0 W respectively it follows that f 0V 0W textstyle f mathbf 0 V mathbf 0 W Let c 0 displaystyle c 0 and v V textstyle mathbf v in V in the equation for homogeneity of degree 1 f 0V f 0v 0f v 0W displaystyle f mathbf 0 V f 0 mathbf v 0f mathbf v mathbf 0 W A linear map V K displaystyle V to K with K displaystyle K viewed as a one dimensional vector space over itself is called a linear functional These statements generalize to any left module RM textstyle R M over a ring R displaystyle R without modification and to any right module upon reversing of the scalar multiplication ExamplesA prototypical example that gives linear maps their name is a function f R R x cx displaystyle f mathbb R to mathbb R x mapsto cx of which the graph is a line through the origin More generally any homothety v cv textstyle mathbf v mapsto c mathbf v centered in the origin of a vector space is a linear map here c is a scalar The zero map x 0 textstyle mathbf x mapsto 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 x2 textstyle x mapsto x 2 is not linear For real numbers the map x x 1 textstyle x mapsto x 1 is not linear but is an affine transformation If A displaystyle A is a m n displaystyle m times n real matrix then A displaystyle A defines a linear map from Rn displaystyle mathbb R n to Rm displaystyle mathbb R m by sending a column vector x Rn displaystyle mathbf x in mathbb R n to the column vector Ax Rm displaystyle A mathbf x in mathbb R m Conversely any linear map between finite dimensional vector spaces can be represented in this manner see the Matrices below If f V W textstyle f V to W is an isometry between real normed spaces such that f 0 0 textstyle f 0 0 then f displaystyle 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 ddx af x bg x adf x dx bdg x dx displaystyle frac d dx left af x bg x right a frac df x dx b frac dg x dx A definite integral over some interval I is a linear map from the space of all real valued integrable functions on I to R displaystyle mathbb R Indeed uv af x bg x dx a uvf x dx b uvg x dx displaystyle int u v left af x bg x right dx a int u v f x dx b 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 displaystyle mathbb R to the space of all real valued differentiable functions on R displaystyle mathbb 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 displaystyle V and W displaystyle W are finite dimensional vector spaces over a field F of respective dimensions m and n then the function that maps linear maps f V W textstyle f V to W to n m matrices in the way described in Matrices 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 displaystyle X and Y displaystyle Y we have E X Y E X E Y displaystyle E X Y E X E Y and E aX aE X displaystyle E aX aE X but the variance of a random variable is not linear The function f R2 R2 textstyle f mathbb R 2 to mathbb R 2 with f x y 2x y textstyle f x y 2x y is a linear map This function scales the x textstyle x component of a vector by the factor 2 textstyle 2 The function f x y 2x y textstyle f x y 2x y is additive It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added f a b f a f b textstyle f mathbf a mathbf b f mathbf a f mathbf b The function f x y 2x y textstyle f x y 2x y is homogeneous It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled f la lf a textstyle f lambda mathbf a lambda f mathbf a Linear extensions Often a linear map is constructed by defining it on a subset of a vector space and then extending by linearity to the linear span of the domain Suppose X displaystyle X and Y displaystyle Y are vector spaces and f S Y displaystyle f S to Y is a function defined on some subset S X displaystyle S subseteq X Then a linear extension of f displaystyle f to X displaystyle X if it exists is a linear map F X Y displaystyle F X to Y defined on X displaystyle X that extends f displaystyle f meaning that F s f s displaystyle F s f s for all s S displaystyle s in S and takes its values from the codomain of f displaystyle f When the subset S displaystyle S is a vector subspace of X displaystyle X then a Y displaystyle Y valued linear extension of f displaystyle f to all of X displaystyle X is guaranteed to exist if and only if f S Y displaystyle f S to Y is a linear map In particular if f displaystyle f has a linear extension to span S displaystyle operatorname span S then it has a linear extension to all of X displaystyle X The map f S Y displaystyle f S to Y can be extended to a linear map F span S Y displaystyle F operatorname span S to Y if and only if whenever n gt 0 displaystyle n gt 0 is an integer c1 cn displaystyle c 1 ldots c n are scalars and s1 sn S displaystyle s 1 ldots s n in S are vectors such that 0 c1s1 cnsn displaystyle 0 c 1 s 1 cdots c n s n then necessarily 0 c1f s1 cnf sn displaystyle 0 c 1 f left s 1 right cdots c n f left s n right If a linear extension of f S Y displaystyle f S to Y exists then the linear extension F span S Y displaystyle F operatorname span S to Y is unique and F c1s1 cnsn c1f s1 cnf sn displaystyle F left c 1 s 1 cdots c n s n right c 1 f left s 1 right cdots c n f left s n right holds for all n c1 cn displaystyle n c 1 ldots c n and s1 sn displaystyle s 1 ldots s n as above If S displaystyle S is linearly independent then every function f S Y displaystyle f S to Y into any vector space has a linear extension to a linear map span S Y displaystyle operatorname span S to Y the converse is also true For example if X R2 displaystyle X mathbb R 2 and Y R displaystyle Y mathbb R then the assignment 1 0 1 displaystyle 1 0 to 1 and 0 1 2 displaystyle 0 1 to 2 can be linearly extended from the linearly independent set of vectors S 1 0 0 1 displaystyle S 1 0 0 1 to a linear map on span 1 0 0 1 R2 displaystyle operatorname span 1 0 0 1 mathbb R 2 The unique linear extension F R2 R displaystyle F mathbb R 2 to mathbb R is the map that sends x y x 1 0 y 0 1 R2 displaystyle x y x 1 0 y 0 1 in mathbb R 2 to F x y x 1 y 2 x 2y displaystyle F x y x 1 y 2 x 2y Every scalar valued linear functional f displaystyle f defined on a vector subspace of a real or complex vector space X displaystyle X has a linear extension to all of X displaystyle X Indeed the Hahn Banach dominated extension theorem even guarantees that when this linear functional f displaystyle f is dominated by some given seminorm p X R displaystyle p X to mathbb R meaning that f m p m displaystyle f m leq p m holds for all m displaystyle m in the domain of f displaystyle f then there exists a linear extension to X displaystyle X that is also dominated by p displaystyle p MatricesIf V displaystyle V and W displaystyle W are finite dimensional vector spaces and a basis is defined for each vector space then every linear map from V displaystyle V to W displaystyle W can be represented by a matrix This is useful because it allows concrete calculations Matrices yield examples of linear maps if A displaystyle A is a real m n displaystyle m times n matrix then f x Ax displaystyle f mathbf x A mathbf x describes a linear map Rn Rm displaystyle mathbb R n to mathbb R m see Euclidean space Let v1 vn displaystyle mathbf v 1 ldots mathbf v n be a basis for V displaystyle V Then every vector v V displaystyle mathbf v in V is uniquely determined by the coefficients c1 cn displaystyle c 1 ldots c n in the field R displaystyle mathbb R v c1v1 cnvn displaystyle mathbf v c 1 mathbf v 1 cdots c n mathbf v n If f V W textstyle f V to W is a linear map f v f c1v1 cnvn c1f v1 cnf vn displaystyle f mathbf v f c 1 mathbf v 1 cdots c n mathbf v n c 1 f mathbf v 1 cdots c n f left mathbf v n right which implies that the function f is entirely determined by the vectors f v1 f vn displaystyle f mathbf v 1 ldots f mathbf v n Now let w1 wm displaystyle mathbf w 1 ldots mathbf w m be a basis for W displaystyle W Then we can represent each vector f vj displaystyle f mathbf v j as f vj a1jw1 amjwm displaystyle f left mathbf v j right a 1j mathbf w 1 cdots a mj mathbf w m Thus the function f displaystyle f is entirely determined by the values of aij displaystyle a ij If we put these values into an m n displaystyle m times n matrix M displaystyle M then we can conveniently use it to compute the vector output of f displaystyle f for any vector in V displaystyle V To get M displaystyle M every column j displaystyle j of M displaystyle M is a vector a1j amj displaystyle begin pmatrix a 1j vdots a mj end pmatrix corresponding to f vj displaystyle f mathbf v j as defined above To define it more clearly for some column j displaystyle j that corresponds to the mapping f vj displaystyle f mathbf v j M a1j amj displaystyle mathbf M begin pmatrix cdots amp a 1j amp cdots amp vdots amp amp a mj amp end pmatrix where M displaystyle M is the matrix of f displaystyle f In other words every column j 1 n displaystyle j 1 ldots n has a corresponding vector f vj displaystyle f mathbf v j whose coordinates a1j amj displaystyle a 1j cdots a mj are the elements of column j displaystyle 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 Matrix for T textstyle T relative to B textstyle B A textstyle A Matrix for T textstyle T relative to B textstyle B A textstyle A Transition matrix from B textstyle B to B textstyle B P textstyle P Transition matrix from B textstyle B to B textstyle B P 1 textstyle P 1 The relationship between matrices in a linear transformation Such that starting in the bottom left corner v B textstyle left mathbf v right B and looking for the bottom right corner T v B textstyle left T left mathbf v right right B one would left multiply that is A v B T v B textstyle A left mathbf v right B left T left mathbf v right right B The equivalent method would be the longer method going clockwise from the same point such that v B textstyle left mathbf v right B is left multiplied with P 1AP textstyle P 1 AP or P 1AP v B T v B textstyle P 1 AP left mathbf v right B left T left mathbf v right right B 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 A 0 110 displaystyle mathbf A begin pmatrix 0 amp 1 1 amp 0 end pmatrix by an angle 8 counterclockwise A cos 8 sin 8sin 8cos 8 displaystyle mathbf A begin pmatrix cos theta amp sin theta sin theta amp cos theta end pmatrix reflection through the x axis A 100 1 displaystyle mathbf A begin pmatrix 1 amp 0 0 amp 1 end pmatrix through the y axis A 1001 displaystyle mathbf A begin pmatrix 1 amp 0 0 amp 1 end pmatrix through a line making an angle 8 with the origin A cos 28sin 28sin 28 cos 28 displaystyle mathbf A begin pmatrix cos 2 theta amp sin 2 theta sin 2 theta amp cos 2 theta end pmatrix scaling by 2 in all directions A 2002 2I displaystyle mathbf A begin pmatrix 2 amp 0 0 amp 2 end pmatrix 2 mathbf I horizontal shear mapping A 1m01 displaystyle mathbf A begin pmatrix 1 amp m 0 amp 1 end pmatrix skew of the y axis by an angle 8 A 1 sin 80cos 8 displaystyle mathbf A begin pmatrix 1 amp sin theta 0 amp cos theta end pmatrix squeeze mapping A k001k displaystyle mathbf A begin pmatrix k amp 0 0 amp frac 1 k end pmatrix projection onto the y axis A 0001 displaystyle mathbf A begin pmatrix 0 amp 0 0 amp 1 end pmatrix 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 mapsThe composition of linear maps is linear if f V W displaystyle f V to W and g W Z textstyle g W to Z are linear then so is their composition g f V Z textstyle g circ f V 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 f1 V W textstyle f 1 V to W and f2 V W textstyle f 2 V to W are linear then so is their pointwise sum f1 f2 displaystyle f 1 f 2 which is defined by f1 f2 x f1 x f2 x displaystyle f 1 f 2 mathbf x f 1 mathbf x f 2 mathbf x If f V W textstyle f V to W is linear and a textstyle alpha is an element of the ground field K textstyle K then the map af textstyle alpha f defined by af x a f x textstyle alpha f mathbf x alpha f mathbf x is also linear Thus the set L V W textstyle mathcal L V W of linear maps from V textstyle V to W textstyle W itself forms a vector space over K textstyle K sometimes denoted Hom V W textstyle operatorname Hom V W Furthermore in the case that V W textstyle V W this vector space denoted End V textstyle operatorname 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 textstyle f V to V is an endomorphism of V textstyle V the set of all such endomorphisms End V textstyle operatorname End V together with addition composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K textstyle K and in particular a ring The multiplicative identity element of this algebra is the identity map id V V textstyle operatorname id V to V An endomorphism of V textstyle V that is also an isomorphism is called an automorphism of V textstyle V The composition of two automorphisms is again an automorphism and the set of all automorphisms of V textstyle V forms a group the automorphism group of V textstyle V which is denoted by Aut V textstyle operatorname Aut V or GL V textstyle operatorname GL V Since the automorphisms are precisely those endomorphisms which possess inverses under composition Aut V textstyle operatorname Aut V is the group of units in the ring End V textstyle operatorname End V If V textstyle V has finite dimension n textstyle n then End V textstyle operatorname End V is isomorphic to the associative algebra of all n n textstyle n times n matrices with entries in K textstyle K The automorphism group of V textstyle V is isomorphic to the general linear group GL n K textstyle operatorname GL n K of all n n textstyle n times n invertible matrices with entries in K textstyle K Kernel image and the rank nullity theoremIf f V W textstyle f V to W is linear we define the kernel and the image or range of f textstyle f by ker f x V f x 0 im f w W w f x x V displaystyle begin aligned ker f amp mathbf x in V f mathbf x mathbf 0 operatorname im f amp mathbf w in W mathbf w f mathbf x mathbf x in V end aligned ker f textstyle ker f is a subspace of V textstyle V and im f textstyle operatorname im f is a subspace of W textstyle W The following dimension formula is known as the rank nullity theorem dim ker f dim im f dim V displaystyle dim ker f dim operatorname im f dim V The number dim im f textstyle dim operatorname im f is also called the rank of f textstyle f and written as rank f textstyle operatorname rank f or sometimes r f textstyle rho f the number dim ker f textstyle dim ker f is called the nullity of f textstyle f and written as null f textstyle operatorname null f or n f textstyle nu f If V textstyle V and W textstyle W are finite dimensional bases have been chosen and f textstyle f is represented by the matrix A textstyle A then the rank and nullity of f textstyle f are equal to the rank and nullity of the matrix A textstyle A respectively CokernelA subtler invariant of a linear transformation f V W textstyle f V to W is the cokernel which is defined as coker f W f V W im f displaystyle operatorname coker f W f V W operatorname im 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 ker f V W coker f 0 displaystyle 0 to ker f to V to W to operatorname coker f 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 lt V the value of x is the freedom in a solution while the cokernel may be expressed via the map W R a b a textstyle a b 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 an bn textstyle left a n right mapsto left b n right with b1 0 and bn 1 an for n gt 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 ℵ0 0 ℵ0 1 textstyle aleph 0 0 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 an cn textstyle left a n right mapsto left c n right with cn an 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 ind f dim ker f dim coker f displaystyle operatorname ind f dim ker f dim operatorname coker 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 Algebraic classifications of linear transformationsNo 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 V and W denote vector spaces over a field F and let T V W be a linear map Monomorphism 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 0V dim 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 Epimorphism 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 0W 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 W Isomorphism 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 T2 T then T is said to be idempotent If T kI where k is some scalar then T is said to be a scaling transformation or scalar multiplication map see scalar matrix Change of basisGiven 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 v AB u displaystyle B left v right AB left u right hence v B 1AB u A u displaystyle left v right B 1 AB left u right A left u 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 ContinuityA 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 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 sin nx n converges to 0 but its derivative cos nx does not so differentiation is not continuous at 0 and by a variation of this argument it is not continuous anywhere ApplicationsA 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 alsoWikibooks has a book on the topic of Linear Algebra Linear Transformations Additive map Z module homomorphism Antilinear map Conjugate homogeneous additive map Bent function Special type of Boolean function Bounded operator Linear transformation between topological vector spaces Cauchy s functional equation Functional equation Continuous linear operator Linear functional Linear map from a vector space to its field of scalarsPages displaying short descriptions of redirect targets Linear isometry Distance preserving mathematical transformationPages displaying short descriptions of redirect targets Category of matrices QuasilinearizationNotes Linear transformations of V into V are often called linear operators on V Rudin 1976 p 207 Let V and W be two real vector spaces A mapping a from V into W Is called a linear mapping or linear transformation or linear operator from V into W if a u v au av textstyle a mathbf u mathbf v a mathbf u a mathbf v for all u v V textstyle mathbf u mathbf v in V a lu lau textstyle a lambda mathbf u lambda a mathbf u for all u V displaystyle mathbf u in V and all real l Bronshtein amp Semendyayev 2004 p 316 Rudin 1991 p 14 Here are some properties of linear mappings L X Y textstyle Lambda X to Y whose proofs are so easy that we omit them it is assumed that A X textstyle A subset X and B Y textstyle B subset Y L0 0 textstyle Lambda 0 0 If A is a subspace or a convex set or a balanced set the same is true of L A textstyle Lambda A If B is a subspace or a convex set or a balanced set the same is true of L 1 B textstyle Lambda 1 B In particular the set L 1 0 x X Lx 0 N L displaystyle Lambda 1 0 mathbf x in X Lambda mathbf x 0 N Lambda is a subspace of X called the null space of L textstyle Lambda Rudin 1991 p 14 Suppose now that X and Y are vector spaces over the same scalar field A mapping L X Y textstyle Lambda X to Y is said to be linear if L ax by aLx bLy textstyle Lambda alpha mathbf x beta mathbf y alpha Lambda mathbf x beta Lambda mathbf y for all x y X textstyle mathbf x mathbf y in X and all scalars a textstyle alpha and b textstyle beta Note that one often writes Lx textstyle Lambda mathbf x rather than L x textstyle Lambda mathbf x when L textstyle Lambda is linear Rudin 1976 p 206 A mapping A of a vector space X into a vector space Y is said to be a linear transformation if A x1 x2 Ax1 Ax2 A cx cAx textstyle A left mathbf x 1 mathbf x 2 right A mathbf x 1 A mathbf x 2 A c mathbf x cA mathbf x for all x x1 x2 X textstyle mathbf x mathbf x 1 mathbf x 2 in X and all scalars c Note that one often writes Ax textstyle A mathbf x instead of A x textstyle A mathbf x if A is linear Rudin 1991 p 14 Linear mappings of X onto its scalar field are called linear functionals terminology What does linear mean in Linear Algebra Mathematics Stack Exchange Retrieved 2021 02 17 Wilansky 2013 pp 21 26 Kubrusly 2001 p 57 Schechter 1996 pp 277 280 Rudin 1976 p 210 Suppose x1 xn textstyle left mathbf x 1 ldots mathbf x n right and y1 ym textstyle left mathbf y 1 ldots mathbf y m right are bases of vector spaces X and Y respectively Then every A L X Y textstyle A in L X Y determines a set of numbers ai j textstyle a i j such that Axj i 1mai jyi 1 j n displaystyle A mathbf x j sum i 1 m a i j mathbf y i quad 1 leq j leq n It is convenient to represent these numbers in a rectangular array of m rows and n columns called an m by n matrix A a1 1a1 2 a1 na2 1a2 2 a2 n am 1am 2 am n displaystyle A begin bmatrix a 1 1 amp a 1 2 amp ldots amp a 1 n a 2 1 amp a 2 2 amp ldots amp a 2 n vdots amp vdots amp ddots amp vdots a m 1 amp a m 2 amp ldots amp a m n end bmatrix Observe that the coordinates ai j textstyle a i j of the vector Axj textstyle A mathbf x j with respect to the basis y1 ym textstyle mathbf y 1 ldots mathbf y m appear in the jth column of A textstyle A The vectors Axj textstyle A mathbf x j are therefore sometimes called the column vectors of A textstyle A With this terminology the range of A is spanned by the column vectors of A textstyle A Axler 2015 p 52 3 3 Tu 2011 p 19 3 1 Horn amp Johnson 2013 0 2 3 Vector spaces associated with a matrix or linear transformation p 6 Katznelson amp Katznelson 2008 p 52 2 5 1 Halmos 1974 p 90 50 Nistor Victor 2001 1994 Index theory Encyclopedia of Mathematics EMS Press 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 Rudin 1991 p 15 1 18 Theorem Let L textstyle Lambda be a linear functional on a topological vector space X Assume Lx 0 textstyle Lambda mathbf x neq 0 for some x X textstyle mathbf x in X Then each of the following four properties implies the other three L textstyle Lambda is continuousThe null space N L textstyle N Lambda is closed N L textstyle N Lambda is not dense in X L textstyle Lambda is bounded in some neighbourhood V of 0 One map F displaystyle F is said to extend another map f displaystyle f if when f displaystyle f is defined at a point s displaystyle s then so is F displaystyle F and F s f s displaystyle F s f s BibliographyAxler Sheldon Jay 2015 Linear Algebra Done Right 3rd ed Springer ISBN 978 3 319 11079 0 Bronshtein I N Semendyayev K A 2004 Handbook of Mathematics 4th ed New York Springer Verlag ISBN 3 540 43491 7 Halmos Paul Richard 1974 1958 Finite Dimensional Vector Spaces 2nd ed Springer ISBN 0 387 90093 4 Horn Roger A Johnson Charles R 2013 Matrix Analysis Second ed Cambridge University Press ISBN 978 0 521 83940 2 Katznelson Yitzhak Katznelson Yonatan R 2008 A Terse Introduction to Linear Algebra American Mathematical Society ISBN 978 0 8218 4419 9 Kubrusly Carlos 2001 Elements of operator theory Boston Birkhauser ISBN 978 1 4757 3328 0 OCLC 754555941 Lang Serge 1987 Linear Algebra Third ed New York Springer Verlag ISBN 0 387 96412 6 Rudin Walter 1973 Functional Analysis International Series in Pure and Applied Mathematics Vol 25 First ed New York NY McGraw Hill Science Engineering Math ISBN 9780070542259 Rudin Walter 1976 Principles of Mathematical Analysis Walter Rudin Student Series in Advanced Mathematics 3rd ed New York McGraw Hill ISBN 978 0 07 054235 8 Rudin Walter 1991 Functional Analysis International Series in Pure and Applied Mathematics Vol 8 Second ed New York NY McGraw Hill Science Engineering Math ISBN 978 0 07 054236 5 OCLC 21163277 Schaefer Helmut H Wolff Manfred P 1999 Topological Vector Spaces GTM Vol 8 Second ed New York NY Springer New York Imprint Springer ISBN 978 1 4612 7155 0 OCLC 840278135 Schechter Eric 1996 Handbook of Analysis and Its Foundations San Diego CA Academic Press ISBN 978 0 12 622760 4 OCLC 175294365 Swartz Charles 1992 An introduction to Functional Analysis New York M Dekker ISBN 978 0 8247 8643 4 OCLC 24909067 Tu Loring W 2011 An Introduction to Manifolds 2nd ed Springer ISBN 978 0 8218 4419 9 Wilansky Albert 2013 Modern Methods in Topological Vector Spaces Mineola New York Dover Publications Inc ISBN 978 0 486 49353 4 OCLC 849801114