linjectif

Linjectif is a rarely used term in mathematics, sometimes employed as a shorthand for a linear injective map. In standard French mathematical language, such a map is more properly described as une application linéaire injective or une fonction linéaire injective.

Definition and meaning: An application f: V → W between vector spaces over a field F is called linjectif when it is one-to-one. Equivalently, its kernel is {0}. By the rank-nullity theorem, dim(Im f) + dim(Ker f) = dim(V); for a linjectif, dim(Im f) = dim(V), hence dim(V) ≤ dim(W).

Properties: The composition of linjectifs is linjectif, when the domains line up appropriately. The inverse of a linjectif, defined on its image, is a linear map from Im(f) to V, but the inverse need not exist as a map from W to V. The image of a linjectif is a subspace of W that is isomorphic to V.

Examples: The inclusion i: R^2 → R^3 given by i(x, y) = (x, y, 0) is linear and injective, and is often treated as a linjectif in informal discussions. More generally, a matrix A with full column rank defines a linjectif f(x) = Ax from F^n to F^m with rank n (n ≤ m).

See also: injection, linear map, rank-nullity theorem.

Terminology: The term linjectif is nonstandard and usage varies; most texts prefer "application linéaire injective" to avoid ambiguity.