Inmathematics, and more specifically inlinear algebra, alinear map (also called alinear mapping,linear transformation,vector space homomorphism, or in some contextslinear function) is amapping${\displaystyle V\to W}$ between twovector spaces that preserves the operations ofvector addition andscalar multiplication. The same names and the same definition are also used for the more general case ofmodules over aring; seeModule homomorphism.

If a linear map is abijection then it is called alinear isomorphism. In the case where${\displaystyle V=W}$, a linear map is called alinear endomorphism. Sometimes the termlinear operator 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${\displaystyle V}$ and${\displaystyle W}$ arereal vector spaces (not necessarily with${\displaystyle V=W}$),^{[citation needed]} or it can be used to emphasize that${\displaystyle V}$ is afunction space, which is a common convention infunctional analysis.^[2] Sometimes the termlinear function has the same meaning aslinear map, while inanalysis it does not.

A linear map from${\displaystyle V}$ to${\displaystyle W}$ always maps the origin of${\displaystyle V}$ to the origin of${\displaystyle W}$. Moreover, it mapslinear subspaces in${\displaystyle V}$ onto linear subspaces in${\displaystyle W}$ (possibly of a lowerdimension);^[3] for example, it maps aplane through theorigin in${\displaystyle V}$ to either a plane through the origin in${\displaystyle W}$, aline through the origin in${\displaystyle W}$, or just the origin in${\displaystyle W}$. Linear maps can often be represented asmatrices, and simple examples includerotation and reflection linear transformations.

In the language ofcategory theory, linear maps are themorphisms of vector spaces, and they form a categoryequivalent tothe one of matrices.

Let${\displaystyle V}$  and${\displaystyle W}$  be vector spaces over the samefield${\displaystyle K}$ . Afunction${\displaystyle f:V\to W}$  is said to be alinear map if for any two vectors$\mathbf{u},\mathbf{v}\in V$  and any scalar${\displaystyle c\in K}$  the following two conditions are satisfied:

• Additivity / operation of addition$${\displaystyle f(\mathbf{u}+\mathbf{v})=f(\mathbf{u})+f(\mathbf{v})}$$ 
• Homogeneity of degree 1 / operation of scalar multiplication$${\displaystyle f(c\mathbf{u})=cf(\mathbf{u})}$$ 

Thus, a linear map is said to beoperation 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.

Bythe associativity of the addition operation denoted as +, for any vectors${\mathbf{u}}_1,\dots ,{\mathbf{u}}_n\in V$  and scalars$c_1,\dots ,c_n\in K,$  the following equality holds:^[4][5]$${\displaystyle f(c_1{\mathbf{u}}_1+\cdots +c_n{\mathbf{u}}_n)=c_{1}f({\mathbf{u}}_1)+\cdots +c_{n}f({\mathbf{u}}_n).}$$  Thus a linear map is one which preserveslinear combinations.

Denoting the zero elements of the vector spaces${\displaystyle V}$  and${\displaystyle W}$  by${\mathbf{0}}_V$  and${\mathbf{0}}_W$  respectively, it follows that$f({\mathbf{0}}_V)={\mathbf{0}}_W.$  Let${\displaystyle c=0}$  and$\mathbf{v}\in V$  in the equation for homogeneity of degree 1:$${\displaystyle f({\mathbf{0}}_V)=f(0\mathbf{v})=0f(\mathbf{v})={\mathbf{0}}_W.}$$ 

A linear map${\displaystyle V\to K}$  with${\displaystyle K}$  viewed as a one-dimensional vector space over itself is called alinear functional.^[6]

These statements generalize to any left-module${}_{R}M$  over a ring${\displaystyle R}$  without modification, and to any right-module upon reversing of the scalar multiplication.

• A prototypical example that gives linear maps their name is a function${\displaystyle f:\mathbb{R}\to \mathbb{R}:x\mapsto cx}$ , of which thegraph is a line through the origin.^[7]
• More generally, anyhomothety$\mathbf{v}\mapsto c\mathbf{v}$  centered in the origin of a vector space is a linear map (herec is a scalar).
• The zero map$\mathbf{x}\mapsto \mathbf{0}$  between two vector spaces (over the samefield) is linear.
• Theidentity map on any module is a linear operator.
• For real numbers, the map$x\mapsto x^2$  is not linear.
• For real numbers, the map$x\mapsto x+1$  is not linear (but is anaffine transformation).
• If${\displaystyle A}$  is a${\displaystyle m\times n}$ real matrix, then${\displaystyle A}$  defines a linear map from${\displaystyle {\mathbb{R}}^n}$  to${\displaystyle {\mathbb{R}}^m}$  by sending acolumn vector${\displaystyle \mathbf{x}\in {\mathbb{R}}^n}$  to the column vector${\displaystyle A\mathbf{x}\in {\mathbb{R}}^m}$ . Conversely, any linear map betweenfinite-dimensional vector spaces can be represented in this manner; see the§ Matrices, below.
• If$f:V\to W$  is anisometry between realnormed spaces such that$f(0)=0$  then${\displaystyle f}$  is a linear map. This result is not necessarily true for complex normed space.^[8]
• Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines alinear operator on the space of allsmooth functions (a linear operator is alinear endomorphism, that is, a linear map with the samedomain andcodomain). Indeed,$${\displaystyle \frac{d}{dx}\left(af(x)+bg(x)\right)=a\frac{df(x)}{dx}+b\frac{dg(x)}{dx}.}$$ 
• A definiteintegral over someintervalI is a linear map from the space of all real-valued integrable functions onI to${\displaystyle \mathbb{R}}$ . Indeed,$${\displaystyle {\int }_u^v\left(af(x)+bg(x)\right)dx=a{\int }_u^{v}f(x)dx+b{\int }_u^{v}g(x)dx.}$$ 
• An indefiniteintegral (orantiderivative) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on${\displaystyle \mathbb{R}}$  to the space of all real-valued, differentiable functions on${\displaystyle \mathbb{R}}$ . Without a fixed starting point, the antiderivative maps to thequotient space of the differentiable functions by the linear space of constant functions.
• If${\displaystyle V}$  and${\displaystyle W}$  are finite-dimensional vector spaces over a fieldF, of respective dimensionsm andn, then the function that maps linear maps$f:V\to W$  ton ×m matrices in the way described in§ Matrices (below) is a linear map, and even alinear isomorphism.
• Theexpected value of arandom variable (which is in fact a function, and as such an element of a vector space) is linear, as for random variables${\displaystyle X}$  and${\displaystyle Y}$  we have${\displaystyle E[X+Y]=E[X]+E[Y]}$  and${\displaystyle E[aX]=aE[X]}$ , but thevariance of a random variable is not linear.

•  
  The function$f:{\mathbb{R}}^2\to {\mathbb{R}}^2$  with$f(x,y)=(2x,y)$  is a linear map. This function scales the$x$  component of a vector by the factor$2$ .
•  
  The function$f(x,y)=(2x,y)$  is additive: It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added:$f(\mathbf{a}+\mathbf{b})=f(\mathbf{a})+f(\mathbf{b})$ 
•  
  The function$f(x,y)=(2x,y)$  is homogeneous: It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled:$f(\lambda \mathbf{a})=\lambda f(\mathbf{a})$ 

Often, a linear map is constructed by defining it on a subset of a vector space and thenextending by linearity to thelinear span of the domain. Suppose${\displaystyle X}$  and${\displaystyle Y}$  are vector spaces and${\displaystyle f:S\to Y}$  is afunction defined on some subset${\displaystyle S\subseteq X.}$  Then alinear extension of${\displaystyle f}$  to${\displaystyle X,}$  if it exists, is a linear map${\displaystyle F:X\to Y}$  defined on${\displaystyle X}$  thatextends${\displaystyle f}$ ^{[note 1]} (meaning that${\displaystyle F(s)=f(s)}$  for all${\displaystyle s\in S}$ ) and takes its values from the codomain of${\displaystyle f.}$ ^[9] When the subset${\displaystyle S}$  is a vector subspace of${\displaystyle X}$  then a (${\displaystyle Y}$ -valued) linear extension of${\displaystyle f}$  to all of${\displaystyle X}$  is guaranteed to exist if (and only if)${\displaystyle f:S\to Y}$  is a linear map.^[9] In particular, if${\displaystyle f}$  has a linear extension to${\displaystyle \mathrm{span}S,}$  then it has a linear extension to all of${\displaystyle X.}$ 

The map${\displaystyle f:S\to Y}$  can be extended to a linear map${\displaystyle F:\mathrm{span}S\to Y}$  if and only if whenever${\displaystyle n>0}$  is an integer,${\displaystyle c_1,\dots ,c_n}$  are scalars, and${\displaystyle s_1,\dots ,s_n\in S}$  are vectors such that${\displaystyle 0=c_1{s}_1+\cdots +c_n{s}_n,}$  then necessarily${\displaystyle 0=c_{1}f\left(s_1\right)+\cdots +c_{n}f\left(s_n\right).}$ ^[10] If a linear extension of${\displaystyle f:S\to Y}$  exists then the linear extension${\displaystyle F:\mathrm{span}S\to Y}$  is unique and$${\displaystyle F\left(c_1{s}_1+\cdots c_n{s}_n\right)=c_{1}f\left(s_1\right)+\cdots +c_{n}f\left(s_n\right)}$$ holds for all${\displaystyle n,c_1,\dots ,c_n,}$  and${\displaystyle s_1,\dots ,s_n}$  as above.^[10] If${\displaystyle S}$  is linearly independent then every function${\displaystyle f:S\to Y}$  into any vector space has a linear extension to a (linear) map${\displaystyle \mathrm{span}S\to Y}$  (the converse is also true).

For example, if${\displaystyle X={\mathbb{R}}^2}$  and${\displaystyle Y=\mathbb{R}}$  then the assignment${\displaystyle (1,0)\to -1}$  and${\displaystyle (0,1)\to 2}$  can be linearly extended from the linearly independent set of vectors${\displaystyle S:=\{(1,0),(0,1)\}}$  to a linear map on${\displaystyle \mathrm{span}\{(1,0),(0,1)\}={\mathbb{R}}^2.}$  The unique linear extension${\displaystyle F:{\mathbb{R}}^2\to \mathbb{R}}$  is the map that sends${\displaystyle (x,y)=x(1,0)+y(0,1)\in {\mathbb{R}}^2}$  to$${\displaystyle F(x,y)=x(-1)+y(2)=-x+2y.}$$ 

Every (scalar-valued)linear functional${\displaystyle f}$  defined on avector subspace of a real or complex vector space${\displaystyle X}$  has a linear extension to all of${\displaystyle X.}$  Indeed, theHahn–Banach dominated extension theorem even guarantees that when this linear functional${\displaystyle f}$  is dominated by some givenseminorm${\displaystyle p:X\to \mathbb{R}}$  (meaning that${\displaystyle |f(m)|\le p(m)}$  holds for all${\displaystyle m}$  in the domain of${\displaystyle f}$ ) then there exists a linear extension to${\displaystyle X}$  that is also dominated by${\displaystyle p.}$ 

If${\displaystyle V}$  and${\displaystyle W}$  arefinite-dimensional vector spaces and abasis is defined for each vector space, then every linear map from${\displaystyle V}$  to${\displaystyle W}$  can be represented by amatrix.^[11] This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if${\displaystyle A}$  is a real${\displaystyle m\times n}$  matrix, then${\displaystyle f(\mathbf{x})=A\mathbf{x}}$  describes a linear map${\displaystyle {\mathbb{R}}^n\to {\mathbb{R}}^m}$  (seeEuclidean space).

Let${\displaystyle \{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}}$  be a basis for${\displaystyle V}$ . Then every vector${\displaystyle \mathbf{v}\in V}$  is uniquely determined by the coefficients${\displaystyle c_1,\dots ,c_n}$  in the field${\displaystyle \mathbb{R}}$ :$${\displaystyle \mathbf{v}=c_1{\mathbf{v}}_1+\cdots +c_n{\mathbf{v}}_n.}$$ 

If$f:V\to W$  is a linear map,$${\displaystyle f(\mathbf{v})=f(c_1{\mathbf{v}}_1+\cdots +c_n{\mathbf{v}}_n)=c_{1}f({\mathbf{v}}_1)+\cdots +c_{n}f\left({\mathbf{v}}_n\right),}$$ 

which implies that the functionf is entirely determined by the vectors${\displaystyle f({\mathbf{v}}_1),\dots ,f({\mathbf{v}}_n)}$ . Now let${\displaystyle \{{\mathbf{w}}_1,\dots ,{\mathbf{w}}_m\}}$  be a basis for${\displaystyle W}$ . Then we can represent each vector${\displaystyle f({\mathbf{v}}_j)}$  as$${\displaystyle f\left({\mathbf{v}}_j\right)=a_{1j}{\mathbf{w}}_1+\cdots +a_{mj}{\mathbf{w}}_m.}$$ 

Thus, the function${\displaystyle f}$  is entirely determined by the values of${\displaystyle a_{ij}}$ . If we put these values into an${\displaystyle m\times n}$  matrix${\displaystyle M}$ , then we can conveniently use it to compute the vector output of${\displaystyle f}$  for any vector in${\displaystyle V}$ . To get${\displaystyle M}$ , every column${\displaystyle j}$  of${\displaystyle M}$  is a vector$${\displaystyle \left(\begin{array}{c}{a}_{1j}\\ ⋮\\ a_{mj}\end{array}\right)}$$ corresponding to${\displaystyle f({\mathbf{v}}_j)}$  as defined above. To define it more clearly, for some column${\displaystyle j}$  that corresponds to the mapping${\displaystyle f({\mathbf{v}}_j)}$ , where${\displaystyle M}$  is the matrix of${\displaystyle f}$ . In other words, every column${\displaystyle j=1,\dots ,n}$  has a corresponding vector${\displaystyle f({\mathbf{v}}_j)}$  whose coordinates${\displaystyle a_{1j},\cdots ,a_{mj}}$  are the elements of column${\displaystyle j}$ . 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:

1. Matrix for$T$  relative to$B$ :$A$ 
2. Matrix for$T$  relative to$B^{\prime }$ :$A^{\prime }$ 
3. Transition matrix from$B^{\prime }$  to$B$ :$P$ 
4. Transition matrix from$B$  to$B^{\prime }$ :$P^{-1}$ 

Such that starting in the bottom left corner${\left[\mathbf{v}\right]}_{B^{\prime }}$  and looking for the bottom right corner${\left[T\left(\mathbf{v}\right)\right]}_{B^{\prime }}$ , one would left-multiply—that is,$A^{\prime }{\left[\mathbf{v}\right]}_{B^{\prime }}={\left[T\left(\mathbf{v}\right)\right]}_{B^{\prime }}$ . The equivalent method would be the "longer" method going clockwise from the same point such that${\left[\mathbf{v}\right]}_{B^{\prime }}$  is left-multiplied with$P^{-1}AP$ , or$P^{-1}AP{\left[\mathbf{v}\right]}_{B^{\prime }}={\left[T\left(\mathbf{v}\right)\right]}_{B^{\prime }}$ .

In two-dimensional spaceR² linear maps are described by 2 × 2matrices. These are some examples:

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

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

The composition of linear maps is linear: if${\displaystyle f:V\to W}$  and$g:W\to Z$  are linear, then so is theircomposition$g\circ f:V\to Z$ . It follows from this that theclass of all vector spaces over a given fieldK, together withK-linear maps asmorphisms, forms acategory.

Theinverse of a linear map, when defined, is again a linear map.

If$f_1:V\to W$  and$f_2:V\to W$  are linear, then so is theirpointwise sum${\displaystyle f_1+f_2}$ , which is defined by${\displaystyle (f_1+f_2)(\mathbf{x})=f_1(\mathbf{x})+f_2(\mathbf{x})}$ .

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

Thus the set$\mathcal{L}(V,W)$  of linear maps from$V$  to$W$  itself forms a vector space over$K$ ,^[12] sometimes denoted$\mathrm{Hom}(V,W)$ .^[13] Furthermore, in the case that$V=W$ , this vector space, denoted$\mathrm{End}(V)$ , is anassociative algebra undercomposition 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 thematrix multiplication, the addition of linear maps corresponds to thematrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

A linear transformation$f:V\to V$  is anendomorphism of$V$ ; the set of all such endomorphisms$\mathrm{End}(V)$  together with addition, composition and scalar multiplication as defined above forms anassociative algebra with identity element over the field$K$  (and in particular aring). The multiplicative identity element of this algebra is theidentity map$\mathrm{id}:V\to V$ .

An endomorphism of$V$  that is also anisomorphism is called anautomorphism of$V$ . The composition of two automorphisms is again an automorphism, and the set of all automorphisms of$V$  forms agroup, theautomorphism group of$V$  which is denoted by$\mathrm{Aut}(V)$  or$\mathrm{GL}(V)$ . Since the automorphisms are precisely thoseendomorphisms which possess inverses under composition,$\mathrm{Aut}(V)$  is the group ofunits in the ring$\mathrm{End}(V)$ .

If$V$  has finite dimension$n$ , then$\mathrm{End}(V)$  isisomorphic to theassociative algebra of all$n\times n$  matrices with entries in$K$ . The automorphism group of$V$  isisomorphic to thegeneral linear group$\mathrm{GL}(n,K)$  of all$n\times n$  invertible matrices with entries in$K$ .

If$f:V\to W$  is linear, we define thekernel and theimage orrange of$f$  by 

$\ker (f)$  is asubspace of$V$  and$\mathrm{im}(f)$  is a subspace of$W$ . The followingdimension formula is known as therank–nullity theorem:^[14]$${\displaystyle \dim (\ker (f))+\dim (\mathrm{im}(f))=\dim (V).}$$ 

The number$\dim (\mathrm{im}(f))$  is also called therank of$f$  and written as$\mathrm{rank}(f)$ , or sometimes,$\rho (f)$ ;^[15][16] the number$\dim (\ker (f))$  is called thenullity of$f$  and written as$\mathrm{null}(f)$  or$\nu (f)$ .^[15][16] 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.

A subtler invariant of a linear transformation$f:V\to W$  is thecokernel, which is defined as$${\displaystyle \mathrm{coker}(f):=W/f(V)=W/\mathrm{im}(f).}$$ 

This is thedual notion to the kernel: just as the kernel is asubspace of thedomain, the co-kernel is aquotient space of thetarget. Formally, one has theexact sequence$${\displaystyle 0\to \ker (f)\to V\to W\to \mathrm{coker}(f)\to 0.}$$ 

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

• the kernel is the space ofsolutions to thehomogeneous equationf(v) = 0, and its dimension is the number ofdegrees of freedom in the space of solutions, if it is not empty;
• the co-kernel is the space ofconstraints 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 spaceW/f(V) is the dimension of the target space minus the dimension of the image.

As a simple example, consider the mapf:R² →R², given byf(x,y) = (0,y). Then for an equationf(x,y) = (a,b) to have a solution, we must havea = 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 ofx is the freedom in a solution – while the cokernel may be expressed via the mapW →R,$(a,b)\mapsto (a)$ : given a vector (a,b), the value ofa is theobstruction to there being a solution.

An example illustrating the infinite-dimensional case is afforded by the mapf:R^∞ →R^∞,$\left\{a_n\right\}\mapsto \left\{b_n\right\}$  withb₁ = 0 andb_{n + 1} =a_n forn > 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 samesum as the rank and the dimension of the co-kernel (${\mathrm{\aleph }}_0+0={\mathrm{\aleph }}_0+1$ ), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of anendomorphism have the same dimension (0 ≠ 1). The reverse situation obtains for the maph:R^∞ →R^∞,$\left\{a_n\right\}\mapsto \left\{c_n\right\}$  withc_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.

For a linear operator with finite-dimensional kernel and co-kernel, one may defineindex as:$${\displaystyle \mathrm{ind}(f):=\dim (\ker (f))-\dim (\mathrm{coker}(f)),}$$ 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 theEuler characteristic of the 2-term complex 0 →V →W → 0. Inoperator theory, the index ofFredholm operators is an object of study, with a major result being theAtiyah–Singer index theorem.^[17]

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.

LetV andW denote vector spaces over a fieldF and letT:V →W be a linear map.

T is said to beinjective or amonomorphism if any of the following equivalent conditions are true:

1. T isone-to-one as a map ofsets.
2. kerT = {0_V}
3. dim(kerT) = 0
4. T ismonic or left-cancellable, which is to say, for any vector spaceU and any pair of linear mapsR:U →V andS:U →V, the equationTR =TS impliesR =S.
5. T isleft-invertible, which is to say there exists a linear mapS:W →V such thatST is theidentity map onV.

T is said to besurjective or anepimorphism if any of the following equivalent conditions are true:

1. T isonto as a map of sets.
2. cokerT = {0_W}
3. T isepic or right-cancellable, which is to say, for any vector spaceU and any pair of linear mapsR:W →U andS:W →U, the equationRT =ST impliesR =S.
4. T isright-invertible, which is to say there exists a linear mapS:W →V such thatTS is theidentity map onW.

T is said to be anisomorphism if it is both left- and right-invertible. This is equivalent toT being both one-to-one and onto (abijection of sets) or also toT being both epic and monic, and so being abimorphism.

IfT:V →V is an endomorphism, then:

• If, for some positive integern, then-th iterate ofT,Tⁿ, is identically zero, thenT is said to benilpotent.
• IfT² =T, thenT is said to beidempotent
• IfT =kI, wherek is some scalar, thenT is said to be a scaling transformation or scalar multiplication map; seescalar matrix.

Given a linear map which is anendomorphism whose matrix isA, in the basisB of the space it transforms vector coordinates [u] as [v] =A[u]. As vectors change with the inverse ofB (vectors coordinates arecontravariant) its inverse transformation is [v] =B[v'].

Substituting this in the first expression$${\displaystyle B\left[v^{\prime }\right]=AB\left[u^{\prime }\right]}$$ hence$${\displaystyle \left[v^{\prime }\right]=B^{-1}AB\left[u^{\prime }\right]=A^{\prime }\left[u^{\prime }\right].}$$ 

Therefore, the matrix in the new basis isA′ =B⁻¹AB, beingB the matrix of the given basis.

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

Alinear transformation betweentopological vector spaces, for examplenormed spaces, may becontinuous. If its domain and codomain are the same, it will then be acontinuous linear operator. A linear operator on a normed linear space is continuous if and only if it isbounded, for example, when the domain is finite-dimensional.^[18] An infinite-dimensional domain may havediscontinuous 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 derivativecos(nx) does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).

A specific application of linear maps is forgeometric transformations, such as those performed incomputer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of atransformation 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 incompiler optimizations of nested-loop code, and inparallelizing compiler techniques.

• 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
• Quasilinearization

• 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.
• 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.