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}
Null Space of a Linear Transformation:-
Let U and V
be two vector spaces over the same field F and let T be a linear transformation
from U into V.
The null
space of T is the set of all those vector α in U such that T(α)=0 (since zero vector of V
The null
space of T is written as N(T) thus
N(T)={α∊U:T(α)=0}
Thus null
space of a linear transformation of T is also called Kernel of T.
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)
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