Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim Null T = 3, dimNullT2 =6. Prove that T does not have a square root; i.e. there does not exist any S ∈ L (V) such that S2 = T.

Give a proof for the standard rule of differentiation, the Chain Rule. To do this, use the following information:

10.1.3 Suppose that the function f is differentiable at c, Then, if f′(c) > 0 and if c is an accumulation point of the set constructed by intersecting the domain of f with (c,∞), then there is a δ > 0 such that at each point xin the domain of f which lies in (c,c+δ) we have f(x) > f(c). If c is an accumulation point of the domain of f intersected with (−∞, c), then there is a δ > 0 such that at each point y in the domain of f which lies in (c−δ,c) we have f(y) < f(c). (Similar case holds for if f′(c) < 0.)

Create two cases for f'(g(x))*g'(x), one where g'(x)=0 (in which case you do nothing), and one where g'(x) not= 0 (in which case you use information from 10.1.3).

Give an orthogonal basis for R3 that contains the vector [1,2,2]T,

1. Identify each of the following statements as true of false and explain your reasoning in full sentences. The domain of discourse is the integers.

(a) ∃x∀y(y > x)

(b) ∀y∃x(y > x)

(c) ∃x(x = 0 → x = 1)

2. Write the formal negation of the following statements. Your negation must not contain any explicit negation symbols.

(a) ∀x∃y(2 < x ≤ y)

(b) ∀y∃x(y > 0 → x ≤ 0)

3. Negate verbally:
(a) ”All people weigh at least 100 pounds.”

(b) ”Somebody did not see any animals in the zoo”

4. If P and Q are predicates over some domain, and if it is true that∀x(P(x)∨ Q(x)), must ∀xP(x)∨∀xQ(x) also be true? Explain.

Locate the results of a recent survey that show at least two variables in a newspaper, magazine, or internet article. Outline the survey data so that your peers can understand the variables and results. Then identify at least one key statistical inference formula involving two of more parameters that you could use to evaluate the data. Provide a brief explanation of why you selected the formula you did and why it matters. Also, explain what the formula is and where it is in the textbook and clearly define your parameters. Do not use regression or correlation examples since we will cover them during the next module.

Two​ drivers, Alison and​ Kevin, are participating in a race. Beginning from a standing​ start, they each proceed with a constant acceleration. Alison covers the last 1/9 of the distance in

7 seconds, whereas Kevin covers the last 1/7 of the distance in 9 seconds. Who wins and by how much​ time?

In Exercises 5–37, solve the given differential equations by Laplace transforms. The function is subject to the given conditions.

13. 4y″ + 4y′ + 5y = 0, y(0) = 1, y′(0) = -1/2

17. y″ + y = 1, y(0) = 1, y′(0) = 1

The Alternate Interior Theorem states that if two lines cut by a transversal line have a pair of congruent alternate interior angles, then they must be parallel. Prove this theorem in Neutral Geometry (first 4 axioms) in two ways

1) using the exterior angle theorem

2) then without.

Hint: use proof by contradiction for both solutions.

Please state these definition and use detal and espilon:

(fn) is a sequence if function:

1.What is the definition of (fn) is continuous?

2.What is the definition of (fn) is uniformly continuous?

3.What is the difference between the function continuous and the sequence of function continuous?

Compute, by Euler’s method, an approximate solution to the following initial value problem for h = 1/8 : y’ = t − y , y(0) = 2 ; y(t) = 3e^(−t) + t − 1 . Find the maximum error over [0, 1] interval.

Summarize seperable equations, integrating factor, exact equations, and auxiliary equation in your own words and what mathematical skills are needed in each technique. Please no handwriting unless I can read it please.

descriptively detail how methods and applications of discrete random variables and continuous random variables are used to support areas of industry research, academic research, and scientific research.