A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. It describes a change compatible with the algebraic structure of vectors, whether those vectors are coordinate lists, polynomials, or functions. Also called a linear map, it is a central object of linear algebra. The domain and target space need not have the same dimension; when they are the same vector space, the transformation is often called a linear operator. (dummit.cos.northeastern.edu)
Definition and basic properties
Let (V) and (W) be vector spaces over the same field (F). A mapping (T:V\to W) is linear if, for every (u,v\in V) and (a,b\in F),
[ T(au+bv)=aT(u)+bT(v). ]
Equivalently, it satisfies both additivity, (T(u+v)=T(u)+T(v)), and homogeneity, (T(av)=aT(v)). These requirements imply (T(0)=0) and preservation of every finite linear combination:
[ T\left(\sum_{i=1}^{k}a_iv_i\right) =\sum_{i=1}^{k}a_iT(v_i). ]
Consequently, a linear transformation is completely determined by its values on a basis of the domain. Conversely, assigning arbitrary target vectors to the members of a basis defines a unique linear transformation. No preservation of lengths, angles, or multiplication of vectors is required. (math.mit.edu)
Matrix representation
For finite-dimensional spaces, choosing ordered bases converts a linear transformation into a matrix. If (V) has dimension (n) and (W) has dimension (m), the representing matrix (A) has (m) rows and (n) columns. Its (j)-th column contains the coordinates of (T(v_j)) in the chosen basis of (W). Thus,
[ [T(v)]_W=A[v]_V, ]
where the brackets denote coordinate columns. In standard coordinates, this becomes (T(x)=Ax). Every (m\times n) matrix defines a linear transformation (F^n\to F^m). (opentext.uleth.ca)
A transformation is an abstract mapping, whereas its matrix depends on the selected bases. For an operator (T:V\to V), changing the basis changes its matrix by similarity:
[ A'=P^{-1}AP, ]
where (P) converts new coordinates into old coordinates. The entries may change, but intrinsic properties such as rank, determinant, and eigenvalues do not. (dummit.cos.northeastern.edu)
Geometric examples and affine maps
In real coordinate spaces, linear transformations include scaling, rotation about the origin, reflection through a subspace containing the origin, shearing, and projection. For example, counterclockwise rotation through angle (\theta) in the plane has matrix
[ R_\theta= \begin{pmatrix} \cos\theta&-\sin\theta\ \sin\theta&\cos\theta \end{pmatrix}, ]
while a horizontal shear has matrix
[ S= \begin{pmatrix} 1&k\ 0&1 \end{pmatrix}. ]
The shear sends ((x,y)) to ((x+ky,y)), changing shape without introducing a translation. An orthogonal projection onto the horizontal axis sends ((x,y)) to ((x,0)), discarding one coordinate. (math.mit.edu)
A linear transformation sends a line to a line or a point, and always fixes the origin. An affine map has the more general form (x\mapsto Ax+b). It is linear exactly when (b=0). Translations therefore are not linear in ordinary coordinates, although homogeneous coordinates allow affine maps to be represented by larger matrices acting linearly on an augmented coordinate space. (groups.csail.mit.edu)
Kernel, image, and rank
Two associated subspaces describe what a transformation loses and what it can produce. Its kernel is
[ \ker T={v\in V:T(v)=0}, ]
and its image is
[ \operatorname{im}T={T(v):v\in V}. ]
Both are linear subspaces, respectively of (V) and (W). The transformation is an injective function exactly when its kernel is ({0}), and a surjective function exactly when its image equals (W). (ximera.osu.edu)
The dimension of the image is the rank; the dimension of the kernel is the nullity. For a finite-dimensional domain, the rank–nullity theorem states
[ \dim V=\dim(\ker T)+\dim(\operatorname{im}T). ]
The rank equals the rank of any representing matrix. Geometrically, nullity counts independent input directions collapsed to zero, while rank counts independent output directions. For (T(x,y,z)=(x,y)), the kernel is the (z)-axis, the image is the entire coordinate plane, and (3=1+2). (ximera.osu.edu)
Composition, inverses, and eigenvectors
The composition of linear transformations is linear. With compatible bases, composition corresponds to matrix multiplication: applying (T) first and then (S) gives matrix (A_SA_T). A bijective linear transformation has a linear inverse and is an isomorphism of vector spaces. For an operator on a finite-dimensional space, invertibility is equivalent to a nonzero determinant of its matrix. (dummit.cos.northeastern.edu)
For an operator (T:V\to V), eigenvalues and eigenvectors identify nonzero vectors whose directions are preserved:
[ T(v)=\lambda v. ]
If a basis consists of eigenvectors, the operator has a diagonal matrix in that basis. Such diagonalization simplifies repeated application, since powers of the diagonal matrix are obtained by raising its diagonal entries to corresponding powers. Not every operator admits an eigenvector basis over its underlying field. (math.uchicago.edu)
Function spaces and continuity
Linearity is not limited to geometric vectors. Differentiation defines a linear transformation on suitable spaces of differentiable functions because
[ D(af+bg)=aDf+bDg. ]
On polynomial spaces, it lowers degree and sends constant polynomials to zero. These examples distinguish linearity in a function-space argument from whether an individual function has a straight-line graph. (dummit.cos.northeastern.edu)
In functional analysis, linear transformations between normed vector spaces require additional consideration of continuity. A linear map is continuous exactly when some finite constant (C) satisfies (|T(v)|\le C|v|) for every (v). Every linear map with finite-dimensional normed domain is continuous, but this need not hold in infinite-dimensional spaces; algebraic linearity alone imposes no such bound. (users.oden.utexas.edu)