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Vector Spaces and Subspaces

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2. Use Theorem 5.2.1 to determine which of the following are subspaces of R3.

Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold.
(a) If u and v are vectors in W, then u + v is in W.
(b) If k is any scalar and u is any vector in W, then ku is in W.

a) all vectors of the form (a, 0, 0)
b) all vectors of the form (a, 1, 1)
c) all vectors of the form (a, b, c), where b = a + c
d) all vectors of the form (a, b, c), where b = a + c + 1
e) all vectors of the form (a, b, 0)

18. Use Theorem 5.2.1 to determine which of the following are subspaces of P3.

Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold.
(c) If u and v are vectors in W, then u + v is in W.
(d) If k is any scalar and u is any vector in W, then ku is in W.

a) all polynomials a0 + a1x +a2x2 + a3x3 for which a0 = 0
b) all polynomials a0 + a1x +a2x2 + a3x3 for which a0 + a1 +a2 + a3 = 0
c) all polynomials a0 + a1x +a2x2 + a3x3 for which a0, a1, a2, a3 are integers
d) all polynomials of the form a0 + a1x, where a0 and a1 are real numbers

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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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• 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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