Prove, that the vector Space Hat (n; F) with the



Answered question


Prove, that the vector Space Hat (n; F) with the multipliсation
AB=ABBA is a F-algebra (algebra over a field F) is such an algbera associative, commutative, untiary?

Answer & Explanation

Joseph Lewis

Joseph Lewis

Beginner2022-01-05Added 43 answers

To prove that the vrctor space Hat(n,f) := {set of n×n matrices over F} is a F-algebra
Note that Hat(n, F) is said to be F-algebra if for any elements x, y, z Hat(n, F) and all elements a, b F.
Right distribulity, left distribulity and compatibility with scalars followed.
Note that
1) (x+y)z=(x+y)zz(x+y)
(Right distribulity)
2) z(x+y)
(Left distribulity)
3) (ax)(by)=axbybyax
(Compatibility with scalars) So, Hat(n,f) is an F-algebra for x, y, z Hat(n, F) and a, b F
Ben Owens

Ben Owens

Beginner2022-01-06Added 27 answers

That is not full answer, here is full:
Note that,
We can check xyx(xy)z
So (Not associative)
Comm? Note that
Not Commutative
Unitary? Mean it sholud have identity element operation *
Let T be identity element
Then xT=x   x H(n, F)
xTTx=x ---- (1)
and Tx=xTxxT=x ---- (2)
from (1) + (2)
So there are no identity elements for all x H(n, f)
Not Unitary

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Commutative Algebra

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?