[SOLVED] Vector spaces over division rings.

Feb 2009
189
0
Hi:
I have the following definition: let V be a vector space over
a division ring D. A mapping a of V into V is called a linear trans-
formation of V if it has the followiwng two properties:
(x+y)a = xa+ya for x,y \(\displaystyle \in\) V,
(x\(\displaystyle \alpha\))a = (xa)\(\displaystyle \alpha\) for x \(\displaystyle \in\) V, \(\displaystyle \alpha\) D

And here I find an odd thing. If a is the mapping multiplication by
a scalar (that is, by an element of D), then a is not in general a linear trans-
formation of V according to the definition, because D needs not be
commutative. Any hint will be welcome.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
First, in English, at least, a set with operations of addition and "scalar multiplication" with the scalars from a division ring rather than a field is usually called a "module", not a "vector space".

Second, If you define scalar multiplication by "\(\displaystyle x\alpha\)" for \(\displaystyle \alpha\) in D, then you must either define \(\displaystyle \alpha x= x\alpha\) or leave \(\displaystyle \alpha x\(\displaystyle undefined. The fact that multiplication of scalars is not commutative has nothing to do with the relationship between \(\displaystyle x\alpha\) and \(\displaystyle \alpha x\).\)\)
 
Nov 2009
485
184
First, in English, at least, a set with operations of addition and "scalar multiplication" with the scalars from a division ring rather than a field is usually called a "module", not a "vector space".
Actually, Thomas Hungerford defines a vector space using division rings.
 

NonCommAlg

MHF Hall of Honor
May 2008
2,295
1,663
Hi:
I have the following definition: let V be a vector space over
a division ring D. A mapping a of V into V is called a linear trans-
formation of V if it has the followiwng two properties:
(x+y)a = xa+ya for x,y \(\displaystyle \in\) V,
(x\(\displaystyle \alpha\))a = (xa)\(\displaystyle \alpha\) for x \(\displaystyle \in\) V, \(\displaystyle \alpha\) D

And here I find an odd thing. If a is the mapping multiplication by
a scalar (that is, by an element of D), then a is not in general a linear trans-
formation of V according to the definition, because D needs not be
commutative. Any hint will be welcome.
multiplication by an element of D is a linear transformation iff that element is in the center of D.
 
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Feb 2009
189
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multiplication by an element of D is a linear transformation iff that element is in the center of D.
Quite understandable. And if D is a field the center of D is D. So in this case that mapping is always a linear transformation. Thanks a lot, NonCommAlg.