lineartransformation

A linear transformation, or linear map, is a function T from a vector space V to a vector space W over a field F that preserves addition and scalar multiplication. That is, for all vectors u and v in V and all scalars c in F, T(u + v) = T(u) + T(v) and T(cu) = c T(u). Linear transformations are the basic maps that respect the algebraic structure of vector spaces.

If V and W are finite‑dimensional and a basis is fixed for each, every linear transformation can be represented by a matrix. Specifically, there exists an m × n matrix A such that T(x) = A x for all x in F^n, where m = dim W and n = dim V. Different choices of bases yield different matrices for the same transformation, related by a change of basis.

Key subspaces associated with a linear transformation are the kernel (or null space) and the image (or range). The kernel consists of all vectors v in V with T(v) = 0, and the image consists of all vectors obtainable as T(v). The rank-nullity theorem states that dim V = rank(T) + nullity(T), where rank(T) is the dimension of the image and nullity(T) is the dimension of the kernel.

An important class of linear transformations are invertible ones. T is invertible if there exists T^{-1}: W → V with T^{-1}∘T = id_V and T∘T^{-1} = id_W. In finite dimensions, invertibility is equivalent to T being bijective, and it implies a one-to-one correspondence between V and W (an isomorphism).

Common examples include the identity map, the zero map, projections, scalings, and rotations in Euclidean spaces. Linear transformations are central in many areas of mathematics and its applications, including computer graphics, differential equations, and data analysis.