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代写data留学生程序 代做R语言程序

核心提示: 代写data留学生程序 代做R语言程序Test 1: Take Home1. Let X denote the set ...

代写data留学生程序 代做R语言程序

Test 1: Take Home

1. Let X denote the set of all irrational numbers x with √

2 ≤ x ≤ 2√2, and

with the usual metric d(x, y) = |x ? y|. Prove that X is not compact.

2. Let (X, d) denote any metric space. The metric space X is called “totally

bounded” when, for every  > 0, there exists finitely many neighborhoods

N(xi) (i = 1, . . . n) such that X ? ∪n

i=1N(xi). The metric space is

“bounded” when { d(x, y) | x, y ∈ R } is a bounded subset of R.

(a) Give an example of a bounded metric space that is not totally bounded.

(b) Prove that every totally bounded metric space is bounded

(c) Prove that a metric space is compact if and only if it is both complete

and totally bounded.

3. Let R

n denote the usual n-dimensional Euclidean space, with its Euclidean

norm

||x|| =vuutXni=1|xi|2

and corresponding metric d(x, y) = ||x ? y||, with x, y ∈ R

n. Given an

n × n matrix T, define

||T|| ≡ sup { ||T x|| | ||x|| ≤ 1 } .

(a) Prove that, for all n × n matrices X and Y , that ||XY || ≤ ||X||||Y||.

(b) Prove that

||T|| = inf { M ∈ R | ||T x|| ≤ M||x|| for all x ∈ Rn}.

(c) With x ∈ R

n, find ||Cx|| when Cx is the n × n matrix with the

coordinates of x in the first column and zeros elsewhere.

(d) With x ∈ R

n, find ||Dx|| when Dx is the n × n diagonal matrix with

the coordinates of x on the main diagonal, and zeros elsewhere.

(e) With x ∈ R

n, find ||Rx|| when Rx is the n × n matrix with the

coordinates of x in the first row and zeros elsewhere.

4. Let T be an n × n matrix, with ||T|| defined as in the previous problem.

Prove that sup { |α| | α an eigenvalue of T }1


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