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Linear Algebra Questions Question 1 Define / mathematically ; the following :" 1 . Define a function which takes the sample mean of a vector ....

Kindly help with the following python conceptual and programming questions:


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Linear Algebra Questions
Question 1 Define / mathematically ; the following :"
1 . Define a function which takes the sample mean of a vector . Hint : use the inner product .
2. Define a function which takes a vector and outputs a difference between a vector and it's associated
Illcall .
3. Define the sample variance and sample covariance functions .

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Question ? All idempatent matrix is one where AA = A. Define P _ *^``TY``". Define MY _ In _ P.
Verify the following for the ordinary least Squares linear model y = *B + {.
1 . P is idempatent*
2. IT is idempatent*
3 . 3 = PU
1 . E = My
5 . V = Py + illy

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Programming Questions
Question & The hailstone sequence is one defined by
I'M =\
In - 1/ ? if In _ I is crell
`I'm - 1 + 1)
otherwise*
Eventually , for natural number values of a] , the hailstone sequence will converge to the cycle { } = 1 , `' * + 1 =
4, Ist ? _ _^its = ] . .... "This is not immediately obvious , but it does happen for all the inputs ] . .... I'll .
What I want you to consider in this question is the generalized hailstone defined by :
I'm - 1/ 2 if In _ I is even!
"In _ 1 + 6
otherwise
where a and & are integer valued in !'O..... 17; . For each of these 121 possible combinations of a and b, Find
out whether or not the sequence converges for all my values 1 . .... Logo . If the sequence does converge for
all these values , find out how many different cycles the sequence converges to . Once you have this dong , try
doing the same thing for the sequence where al , an , boy, big are in 10 . .... 10) :
if an - I mad } = "I'm -
"IIn _ 1 + by if an_ I mad` = 1
again _ It by if an _ I mad } = ?

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