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# Matrices, Vector Spaces and Subspaces

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Give a demonstration as to why or why not the given objects are vector subspaces of M22

a) all 2 X 2 matrices with integer entries

A vector space is a set that is closed under finite vector addition and scalar multiplication.

It is not a vector space, since V is NOT closed under finite scalar multiplication. For instance, take a 2 by 2 matrix with integer entries

Then is NOT in V since not every entry is an integer.

b) all matrices a b

c d such that: a + b + c + d = 0

It is a vector space, since V is closed under finite vector addition and scalar multiplication.

(1) V is closed under finite vector addition

For instance, take any two 2 by 2 matrices in V

and
where and . Obviously, we have
=
as

And
(2) V is closed under finite scalar multiplication.

as

c) all 2 X 2 matrices A such that det(A) = 0

It is NOT a vector space, since V is NOT closed under finite vector addition. For instance, take TWO 2 by 2 matrices with det(A)=det(B)=0

and
BUT
with det(A+B)=2*2-1*1=3
So,

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Matrices, Vector Spaces and Subspaces are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
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• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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