Let B equal the matrix below:

[{1,0,0},{0,3,2},{2,-2,-1}]

(1,0,0 is the first row of the matrix B)

(0,3,2 is the 2nd row of the matrix B

(2,-2,-1 is the third row of the matrix B.)

1) Determine the eigenvalues and associated eigenvectors of B. State both the algebraic and geometric multiplicity of the eigenvalues.

2) The matrix is defective. Nonetheless, find the general solution to the system x’ = Bx. (x is a vector)

4: \textbf{Proof} Prove that if $A$ and $B$ are countable sets, then $A \cup B$ is countable.

5: Use induction and problem 4 to prove that if $A_1, A_2, ..., A_m$ are each countable sets, then the union $A_1 \cup A_2 \cup ... \cup A_m$ is countable.

#5 please

if (a_n) is a real cauchy sequence and b is also real, prove

i) (|a_n|) is a cauchy sequence

ii) (ba_n) is also a cauchy sequence

"•“"Suppose ℱ, ?1, and ?2 are nonempty families of sets. Prove that if ℱ ⊆ ?1 ∩ ?2, then ∩?1 ∪ ∩?2 ⊆ ∩ℱ.“"

Please explain what the question is asking for and break down the solution for me step by step with explanations please!

Use the transformation ,  to express  as an iterated integral in the order dudv, where R is the triangular region with vertices (0,0), (1,4), and (4,1).

Keep your eyes and ears open as you read or listen to the news this week. Find/discover an example of statistics & probability in the news. Was it explained well, or poorly? What is the context? Do you think anything about the article is misleading? Can you make any inferences based on it?

Do you think that the statistics in the article would convince the reader that we are all facing a serious problem with our environment? Explain why or why not by referencing examples from the article?

Solve the initial value problem.

(x^2 * D^2 +xD - 4i) * y = x^3, y(1) = -4/5, y'(1) = 93/5

let A be a real matrix of size M*N.,assume that nullity (A^T* A)=r. Find the range of values that 'r' can take using values of 'M' and 'N'. also find the nullity (A^T)

Problem 3-12

Suppose that you sell short 250 shares of Xtel, currently selling for $100 per share, and give your broker $15,000 to establish your margin account.

a. If you earn no interest on the funds in your margin account, what will be your rate of return after one year if Xtel stock is selling at: (i) $115; (ii) $100; (iii) $95? Assume that Xtel pays no dividends. (Leave no cells blank - be certain to enter "0" wherever required. Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)

b. If the maintenance margin is 25%, how high can Xtel’s price rise before you get a margin call? (Round your answer to 2 decimal places.)

c. Redo parts (a) and (b), but now assume that Xtel also has paid a year-end dividend of $1 per share. The prices in part (a) should be interpreted as ex-dividend, that is, prices after the dividend has been paid. (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)

State the linearity of the equation: x (y^2 + 2z) p − y (x^2 + 2z) q = (x^2 −y^2) z. Find the general solution, and then the particular solution that passes through the line: x + y = 0, z = 2.
Linearity is the category to which it belongs among: linear, quasi-linear or non-linear.

1. Find the Laplace transform of each of the following functions: (a). f(t) = t , (b). f(t) = t²  , (c) f(t) = tⁿ  where n is a positive integer

Laplace transform of the given function

2. . f(t) = sin bt

3.   f(t) = e^at sin bt

By induction:

1. Prove that Σⁿ_i=1(2i − 1) = n²

2. Prove thatΣⁿ_i=1 i² = n(n+1)(2n+1) / 6 .

Describe the set of all unit tight frames with exactly five vectors, up to PRR-equivalence.

1.2.2*. Find an example to show that the phrase "finitely many" is necessary in the statement of Lemma 1.2.3 (iii).

Lemma 1.2.3.

(i) Rn and empty set are open in Rn
(ii) The union ofo pen subset of Rn is open.

(iii) The intersect of finitely many open subsets Rn is open.