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Ines B.
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French
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Question:

Decris un ami proche en deux lignes. (physique, caractere, profession...)

Ines B.

Mon ami s'appelle isabelle, elle est petite de taille, blonde aux yeux bleu. Isabelle est tres intelligente et elle est professeur par profession.

Linear Algebra
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Question:

Show the following: (a) If {v1, . . . , vn} is a spanning set for a vector space V and v is any vector in V , then v, v1, . . . , vn are linearly dependent. (b) If v1, . . . , vn are linearly independent vectors in a vector space V , then v2, . . . , vn cannot span V .

Ines B.

Since {v1, . . . , vn} is a spanning set for V and v is in V , there are scalars α1, . . . , αn such that v = α1v1 + · · · + αnvn . Thus α1v1 + · · · + αnvn − v = 0 , so 0 can be written as a linear combination of v, v1, . . . , vn, with not all coefficients vanishing (the coefficient of v is −1 6= 0), so v, v1, . . . , vn are linearly dependent. (b) By contradiction. Let v1, . . . , vn be linearly independent. Suppose to the contrary that {v2, . . . , vn} is a spanning set for V . Since v1 ∈ V , by (a), the vectors v1, v2, . . . , vn must be linearly dependent, which is a contradiction. So the vectors v2, . . . , vn cannot span V

Statistics
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Question:

A sample of size n is taken from a N (µ, σ2 ) population, with µ known and σ 2 unknown. You wish to test null-hypothesis H0 : σ 2 = σ 2 0 against H1 : σ 2 6= σ 2 0 with significance level α. (a) Suggest a z-test (i.e. with normally distributed test statistic).

Ines B.

(a) Under H0 i.e. when X1, . . . , Xn are iid N (µ, σ2 0 ) we have Z := X¯ − µ σ0/ √ n ∼ N (0, 1). The critical region for the 2-sided test is (−∞, −zα/2) ∪ (zα/2 ,∞).

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