Prove, that the vector Space Hat (n; F) with the multipliсation A⋅B=AB−BA is a F-algebra...

untchick04tm

untchick04tm

Answered

2022-01-04

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

Expert

2022-01-05Added 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)
=xz+yzzxzy
=(xzzx)+(yzzy)
=xz+yz
(Right distribulity)
2) z(x+y)
=z(x+y)(x+y)z
=(zxxz)+(zyyz)
=zx+zy
(Left distribulity)
3) (ax)(by)=axbybyax
=ab(xyyx)
=ab(xy)
(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

Expert

2022-01-06Added 27 answers

That is not full answer, here is full:
Note that,
(xy)z=(xyyx)z
=(xyyz)zz(xyyx)
=xyzyxzzxy+zyx
and
x(yz)=x(yzzy)
=x(yzzy)(yzzy)x
=xyzxzyyzx+zyx
We can check xyx(xy)z
So (Not associative)
Comm? Note that
xy=xyyz
yx=yxxy
xyyx
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)
2x=0x=0
So there are no identity elements for all x H(n, f)
Not Unitary

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