 untchick04tm

2022-01-04

Prove, that the vector Space Hat (n; F) with the multipliсation
$A\cdot B\phantom{\rule{0.222em}{0ex}}=AB-BA$ is a F-algebra (algebra over a field F) is such an algbera associative, commutative, untiary? Joseph Lewis

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 $\in$ Hat(n, F) and all elements a, b $\in$ F.
Right distribulity, left distribulity and compatibility with scalars followed.
Note that
1) $\left(x+y\right)\cdot z=\left(x+y\right)z-z\left(x+y\right)$
$=xz+yz-zx-zy$
$=\left(xz-zx\right)+\left(yz-zy\right)$
$=x\cdot z+y\cdot z$
(Right distribulity)
2) $z\cdot \left(x+y\right)$
$=z\left(x+y\right)-\left(x+y\right)z$
$=\left(zx-xz\right)+\left(zy-yz\right)$
$=z\cdot x+z\cdot y$
(Left distribulity)
3) $\left(ax\right)\cdot \left(by\right)=axby-byax$
$=ab\left(xy-yx\right)$
$=ab\left(x\cdot y\right)$
(Compatibility with scalars) So, Hat(n,f) is an F-algebra for x, y, z $\in$ Hat(n, F) and a, b $\in$ F Ben Owens

That is not full answer, here is full:
Note that,
$\left(x\cdot y\right)\cdot z=\left(xy-yx\right)\cdot z$
$=\left(xy-yz\right)z-z\left(xy-yx\right)$
$=xyz-yxz-zxy+zyx$
and
$x\cdot \left(y\cdot z\right)=x\cdot \left(yz-zy\right)$
$=x\left(yz-zy\right)-\left(yz-zy\right)x$
$=xyz-xzy-yzx+zyx$
We can check $x\cdot y\cdot x\ne \left(x\cdot y\right)\cdot z$
So (Not associative)
Comm? Note that
$x\cdot y=xy-yz$
$y\cdot x=yx-xy$
$\therefore x\cdot y\ne y\cdot x$
$\therefore$ Not Commutative
Unitary? Mean it sholud have identity element operation *
Let T be identity element
Then H(n, F)
$⇒xT-Tx=x$ ---- (1)
and $T\cdot x=x⇒Tx-xT=x$ ---- (2)
$\therefore$ from (1) + (2)
$\to 2x=0⇒x=0$
So there are no identity elements for all x $\in$ H(n, f)
Not Unitary

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