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 (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
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
[parent] left / right perpendicular (Derivation)

Given a sesquilinear form $ b:V\times V\rightarrow k$ over the field $ k$, if $ v,u\in V$ such that $ b(v,u)=0$ then we say $ v$ is right perpendicular to $ u$ and denote it $ v\bot u$. Likewise $ u$ is left perpendicular to $ v$ and can be denoted by $ u\top v$.

By definition $ v\perp u$ if and only if $ u\top v$. However, $ v\perp u$ need not imply $ u\perp v$.

For example, let $ V=k\oplus k$ and $ b( (v_1,v_2), (u_1,u_2))=v_1 u_2$. Then $ b((0,1),(1,0))=0$ so $ (0,1)\bot (1,0)$, or equivalently, $ (1,0)\top (0,1)$. However $ b((1,0),(0,1))=1\neq 0$ so $ (1,0)$ is not right perpendicular to $ (0,1)$ and $ (0,1)$ is not left perpendicular to $ (1,0)$.



"left / right perpendicular" is owned by Algeboy.
(view preamble | get metadata)

View style:

Also defines:  left perpendicular, right perpendicular
Keywords:  perpendicular

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: imply, definition, field, sesquilinear form
There is 1 reference to this entry.

This is version 4 of left / right perpendicular, born on 2006-09-06, modified 2006-09-07.
Object id is 8315, canonical name is LeftRightPerpendicular.
Accessed 1380 times total.

Classification:
AMS MSC: 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

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