image of a linear transformation

Definition Let T:VW be a linear transformation. Then the image of T is the set

Im(T)={wWw=T(v)for somevV}=T(V).

0.0.1 Properties

  1. 1.

    The dimensionPlanetmathPlanetmath of Im(T) is called the rank of T;

  2. 2.

    T is a surjection, if and only if Im(T)=W;

  3. 3.

    Im(T) is a vector subspace of W;

  4. 4.

    If L:WU is a linear transformation, then Im(LT)=L(Im(T));

Title image of a linear transformation
Canonical name ImageOfALinearTransformation
Date of creation 2013-03-22 13:48:32
Last modified on 2013-03-22 13:48:32
Owner Koro (127)
Last modified by Koro (127)
Numerical id 8
Author Koro (127)
Entry type Definition
Classification msc 15A04
Related topic RankNullityTheorem
Related topic KernelOfALinearTransformation