Linear mappings. Hom (V, W). Image, kernel, rank, nullity. Singular and nonsingular mappings.               

Theorem 1. Every linear mapping T: V W from an abstract n-dimensional vector space V to an abstract m-dimensional vector space W can be represented by some mxn matrix A called the matrix representation of the transformation..

The set of all possible linear mappings from an abstract n-dimensional vector space V over a field F to an abstract m-dimensional vector space W over F corresponds to the set of all possible mxn matrices over F.

Hom (V,W). The set consisting of all possible linear mappings from a vector space V to a vector space W is called Hom (V,W). For the case when V and W are both finite dimensional each linear mapping contained in Hom (V,W) corresponds to some matrix. Let V be of dimension n, W of dimension m, and both be over field F. Then the set of all linear mappings of the set Hom (V,W) corresponds to the set of all possible mxn matrices over the field F. A special case of Hom (V,W) is Hom(V,V), the set of all possible mappings of a vector space onto itself. If V is of finite dimension n over field F, Hom (V,V) corresponds to the set of all possible n-square matrices over F.

Isomorphic linear mapping A linear mapping T: V W represents a homomorphism of V into W. If the mapping is one-to-one it represents an isomorphism.. The vector spaces V and W are said to be isomorphic if there is an isomorphism of V onto W.

Image of a linear mapping. The image of a linear mapping T: V W is the set of images in W into which the elements of V map. It is denoted by Im T. The image of a linear mapping is the same as its range (the terms are used synonymously). If V and W are finite dimensional spaces and A is the matrix representation of transformation T then the image of T corresponds to the column space of matrix A.

Kernel of a linear mapping. The kernel of a linear mapping T: V W is the set of elements in the domain V which map into 0 W. It is denoted by Ker T. The term “kernel” is synonymous with the term “null space”. If V and W are finite dimensional spaces and A is the matrix representation of transformation T then the kernel of T corresponds to the null space of matrix A.

Singular and nonsingular mappings. A linear mapping T: V W is said to be singular if it maps some nonzero vector in V into 0 W. If it maps only 0 V into 0 W it is said to be nonsingular. A mapping is inonsingular if and only if it is one-to-one. A nonsingular mapping possesses an inverse; a singular mapping does not. If V and W are finite dimensional spaces and A is the matrix representation of transformation T then the mapping is singular if matrix A is singular, nonsingular if it is not.

Rank of a linear mapping. The rank of a linear mapping T: V W is defined to be the dimension of its image. If V and W are finite dimensional spaces and A is the matrix representation of transformation T the rank of T is the rank of matrix A. [Note. The rank of matrix A is equal to the dimension of its column space and the dimension of its column space is the dimension of the image.].

Nullity of a linear mapping. The nullity of a linear mapping is defined to be the dimension of its kernel (or null space).

Let T: V W be a linear mapping. Then the image of T is a subspace of W and the kernel of T is a subspace of V. If V is of finite dimension then

            dim V = dim (Ker T) + dim (Im T)

That is, the sum of the dimensions of the image and kernel of a linear mapping is equal to the dimension of its domain. Also, because the rank of a linear mapping is equal to the dimension of it image,

            dim V = nullity (T) + rank (T)

Theorem 2. Let V and W be vector spaces over a field F. Let (v₁,v₂, ... ,v_n) be a basis of V and let (w₁,w₂, ... ,w_n) be any vectors in W. Then there exists a unique linear mapping

T: V W such that

            T(v₁) = w₁, T(v₂) = w₂, ......, T(v_n) = w_n

The vectors w₁,w₂, ... ,w_n are completely arbitrary. They may be linearly dependent and they may even be equal to each other.

References

Lipschutz. Linear Algebra