lineartransform

A linear transform, or linear transformation, is a function T: V -> W between vector spaces that preserves vector addition and scalar multiplication: T(u+v) = T(u) + T(v) and T(c v) = c T(v) for all vectors u,v in V and scalars c. If V and W are finite-dimensional and defined over a common field, T can be represented by a matrix relative to chosen bases, so that T(v) = A v.

In Euclidean spaces, typical linear transforms include identity, zero transformation, projections onto subspaces, rotations, scalings, and shears. These maps are completely determined by their action on a basis, and their properties are studied via the matrix A.

The kernel (null space) of T is the set of vectors mapped to zero; the image (range) is the set of all outputs. The dimensions satisfy the rank-nullity theorem: dim V = dim ker T + dim im T. For m x n matrices, rank is at most min(m,n) and equals the dimension of the image.

A linear transform is invertible if there exists a linear transform S with ST = TS = I; this occurs exactly when the representing matrix A is square and invertible. The inverse corresponds to A^{-1}. Eigenvalues and eigenvectors describe vectors that are scaled by T.

Applications span computer graphics, signal processing, data compression, and solving systems of linear equations. In many contexts, linear transforms form the backbone of methods for changing representation, simplifying structures, or projecting data into meaningful subspaces.