Image of a Transformation

Overview

Definitions

• {% Im(T) = \{ \textbf{u} \; s.t. \; T(\textbf{v}) = \textbf{u} \; for \; some \; \textbf{v} \} %}
  The image of T is the set of vectors in the range for which there is an element in the domain that maps to that vector.
• {% ker(T) = \{ \textbf{v} \in V | T(\textbf{v}) = \textbf{0} \} %}
  The kernel is the set of vectors which map to the zero vector

Theorems

• For a given linear transformation {% T:V \rightarrow W %}
  {% dim(V) = rank(T) + nullity(T) %}
  where {% rank(T) = dim(Im(T)) %} and {% nullity(T) = dim(Ker(T)) %}