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MPFG C1

 
 
JinTina的头像
MPFG C1
JinTina - 2024年08月8日 Thursday 21:17
 

Hello,


For problem 12 in the packet, I don't understand the last few steps. After getting the "matrix" of "z"s, how did you get that that is 49? I listed out the terms explicitly, and I got:




and since z^(-k)=z^(7-k) by seeing the roots on the unit circle, I got that its just 7(z^0+z^1+...+z^6), but then I'm just doing a loop because we got before that z^0+z^1+...+z^6=2/(1-z^7).


Additionally, why is this reasoning wrong:

b/c z+z^2+z^3+...+z^6+z^7=0 by the fact that the sum of the roots is 0, z+z^2+z^3+...+z^6=0-z^7=0-(-1)=1, thus 1+z+z^2+z^3+...+z^6=1+1=2, but the teacher got that it equals 2/(1-z^7). (I understand how the teacher got that, but I just don't understand what is wrong with my reasoning)


Thanks!

Tina Jin





 
WangDr. Kevin的头像
Re: MPFG C1
WangDr. Kevin - 2024年08月8日 Thursday 23:20
 

The first part: Can't see what you got there.  Was it an image?  It probably didn't upload.

The second part: since $z^7 = -1$ for this problem, we don't have $z^{-k} = z^{7-k}$.  In fact $z^{-k} = - z^{7-k}$.

Also I don't think I said $1+z+z^2+z^3+z^4+z^5+z^6=\dfrac{2}{1-z^7}$.  It is supposed to be $\dfrac{2}{1-z}$.

The third part, also check that $z^7=-1$ by the problem statement.

JinTina的头像
Re: MPFG C1
JinTina - 2024年08月9日 Friday 18:25
 

Hello Dr. Wang,


First part: 

My apologies for the image not loading; here it is (hopefully it will load this time):


Second part: oops, I get it now, thank you!

Third part: I indeed used z^7=-1, so my reasoning should be correct...but its wrong. What is wrong with my reasoning? (its in the earlier discussion post)


Thank you!

Tina Jin

WangDr. Kevin的头像
Re: MPFG C1
WangDr. Kevin - 2024年08月11日 Sunday 00:35
 

The image you got is correct.  Since $z^7=-1$, the expression for each $z$ is not simplified to $7(z^0+z^1+z^2+\cdots+z^6)$.  What we can do here is to calculate the sum over all roots for each term.  For the term $7z^0$, we are calculating $7z_1^0+7z_2^0+7z_3^0+\cdots +7z_7^0$, here $z_i$ represents the seven 7th roots of $-1$.  This is simply seven $7$s, which sums to 49.

For the other terms, they are all of the form $z^k$ where $k\neq 0$, and we will have

$$z_1^k + z_2^k + \cdots + z_7^k=0,$$

because for a particular fixed $k$, all $z_i^k$ are evenly distributed on the unit circle, since $z_i$ are evenly distributed on the unit circle.  Thus the sum is $0$ for this $k$.

Therefore the total sum is $49$.

For the third part, it is still because $z^7=-1$ instead of $1$.  Thus the terms $z, z^2, z^3, \ldots, z^6$ are not all roots.  Some of them aren't roots of $-1$ (for example, $z^2$ is actually a 7th root of $1$.). Therefore their sum is not $1$.

JinTina的头像
Re: MPFG C1
JinTina - 2024年08月12日 Monday 06:02
 

Hello Dr. Wang,


I get it, thank you!


Tina Jin