Range,Rank,Nullity Of A Linear Transformation

Range of a Linear Transformation:-

Let U and V be two vector spaces over the field F.Let T be a linear transformation from U into V then the of all vectors of V which are images of elements of U is called range of linear transformation. It is denoted by R(T).

Thus R(T)={T(α)∊V:α∊U}

 

Rank of a linear transformation:-

Let T be a linear transformation from a vector space U into a vector space V with U as finite dimensional then the rank of T, is denoted by ρ(T) and is defined as the dimension of range of T.

i.e.                         ρ(T)=dim R(T)

Nullity of a linear transformation:-

Let T be a linear transformation from a vector space U into a vector space V with U as finite dimensional then nullity of linear  transformation is denoted by υ(T) and defined as the dimension of null space of T.

i.e.                       υ(T)=dim N(T)