回忆题求助

PalinaUID：3394315

2010 Q33
T is orthogonal transformation, which of following is correct?
1. Tu*Tv=u*v
2. if v is orthogonal to (1/sqrt(2),1/sqrt(2),0), Tv*v=0
3. T^2=I

2009 Q34
Which one is a common divisor of x^2+x+1 and x^3+x+1
1. x+1
2. x+2
3. x^2+x+1

Answer is 2, could anyone explain the method?

2008 Q53
以下哪个多项式在（or不在）二元域1+x^2和1+x^3生成的理想里
1. 1+x^4
2.
3. 1+x^6

Q57
已知H是G的子群，问以下哪个正确
1.H左陪集个数与右陪集个数一样多
2.若xH不等于Hy,那么这两个集交为空
3.xH=Hx

treemantanUID：2770681

2009 Q34
for abstract algebra, you only need to examine each one. So when considering the common divisor of polynomial, you need to check the root of it. This question lies on the Z_3, the first step is to substitute 0,1,2 to the given poly and then you can find that x=1 is the only root. So x-1 is the common divisor and x+2 can be rewritten to x-1 when minus 3 which is equal to 0 on Z_3.
The rest can be easily examined in the same way.

2008 Q53
Please be careful that this is on the Z_2. So -1=1. And it is to some degree the same as the previous.

Q57.
1. Cosets mean to partition the group. And regardless of what side, the number of Coset is determined by the H because H is a coset.
2.It's right. If xH!=Hy but there exists xg_1=g_2y, because the coset is a equivalence relation. So xH=Hy that is a contradiction. Thus, the intersection of them is empty

3. Wrong. It's not necessarily left=right.  And this form in the abstract Algebra is called center of groups that lead to define the normal of groups.

...orthogonal operator,forgotten. Wiki

treemantanUID：2770681

Wiki is enough.
For the 2010 Q33
the orthogonal transformation is the linear transformation that preserve the inner product.
i.e. <u,v>=<Tu,Tv>.
And a useful example is the SO2 that is the orthogonal matrix group.
Rotation and reflection is the stiff movement that can be represented by the SO2. Then
1.It's definition. Right.
2.Thought v is perpendicular to a vector and let Tv be sort of like a rotation of v and then Tv is not necessarily perpendicular to v itself.
And this is wrong.
3.if T is a rotation of 30 degree. T^2 apparently is not equal to I. And dihedral group can be used as a example.

12345drUID：3394416

Q57第二个是错的吧？比如6阶二面体群中生成元a，b，其中a^3=1,b^2=1,bab=a^2,则其中a<b>和<b>a不相等，但交集就不是空的

treemantanUID：2770681

12345dr 发表于 2012-11-3 00:35
Q57第二个是错的吧？比如6阶二面体群中生成元a，b，其中a^3=1,b^2=1,bab=a^2,则其中a和a不相等，但交集就不 ...

YEah，2 is wrong