PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
image of a linear transformation (Definition)

Definition Let $ T:V\to W$ be a linear transformation. Then the image of $ T$ is the set

$\displaystyle \operatorname{Im} (T) = \{ w\in W \mid w=T(v) \,$for some$\displaystyle \, v\in V\} = T(V).$

Properties

  1. The dimension of $ \operatorname{Im}(T)$ is called the rank of $ T$;
  2. $ T$ is a surjection, if and only if $ \operatorname{Im}(T)=W$;
  3. $ \operatorname{Im}(T)$ is a vector subspace of $ W$;
  4. If $ L\colon W\to U$ is a linear transformation, then $ \operatorname{Im}(LT) =L(\operatorname{Im}(T))$;



"image of a linear transformation" is owned by Koro. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: rank-nullity theorem, kernel of a linear mapping
Log in to rate this entry.
(view current ratings)

Cross-references: vector subspace, surjection, rank, dimension, linear transformation
There is 1 reference to this entry.

This is version 5 of image of a linear transformation, born on 2003-07-29, modified 2004-09-24.
Object id is 4530, canonical name is ImageOfALinearTransformation.
Accessed 4665 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)