Suppose A is an n × n matrix with the property that the equation Ax = b has at least one solution for each b in R n . Explain why each equation Ax = b has in fact exactly one solution

please explain step by step. see if the given matrix A is diagonalizable, if so, give a diagonal matrix similar to A as well as a matrix P so that P⁻¹AP is the diagonal matrix below.

3 1 -1

0 0 -2

0 1 2

maclaurin series for x=.5, 1.5 radians f(x)= (tanx - sinx -2.75) to the third derivative

The numerical aspect of statistics can be described as "numbers with social context." What does this mean to you?

Solve the initial value problem using systems of Linear Differential Equations. Please try to use computer typing

x’ – 4y = 3

x + y’ = 2

IVP: x(0) = 0, y(0) = 1

State and sketch a proof of Cauchy’s Theorem (not Cauchy’s Integral Formula). You need not show all of the details, just describe the general steps.

Calculate the integral from (-infinty) to (+ infinity) of [x^2 / (x^4 + 4)[ dx

The half-life of carbon-14 is 5730 years. Assume that the decay rate is proportional to the amount. Find the age of a sample in which 10% of C-14 originally present have decayed.
Select the correct answer.

Part 4:

• Discuss in one paragraph which aspects of your roller coaster would realistically make sense in the real world and which aspects would not. Make sure you refer to graphs you used in your answer.
• Looking at the example below, explain why your function roller coaster cannot have a loop.

a) State Mellin’s inverse Laplace transform formula.
b) State Cauchy’s residue theorem.
iii. Use (a) and (b) to prove that the inverse Laplace transform of F(s)=1/(s+a) is equal to f(t)= e^(-at),t>0

Find the solution of the following problems. Before doing these problems, you might want to review Exercise 3** on page 63:

d.) xy" + y' = x, where y(1) = 1m and y'(1) = -1 (answer should be y(x) = 1/4 x² - 3/2 ln(x) + 3/4)

e.) (x-1)²y" + (x-1)y' - y = 0, where y(2) = 1, and y'(2) = 0 (answer should be: y(x) = 1/2 (x-1)^-1 + x/2 - 1/2)

**Exercise 3: The formula for a particular solution given in (3.42) applies to the more general problem of solving y" + p(t)y' + q(t)y = f(t). In this case, y₁ and y₂ are independent solutions of the associated homogeneous equation y" + p(t)y' + q(t)y = 0.

Please show work!

True or False.

1. If the set {v1,v2} is a basis of R^2, then the set {v1,v1+v2} is also a basis of R^2.

2.If W be a vector space and V1,V2 are subspaces of W, then V1 u V2 is also a subspace of W. V1 u V2 denotes the union of V1 and V2, i.e. the set of vectors which belong to either V1 or V2 (or to both).

3.If W be a vector space and V1,V2 are subspaces of W, then V1 ^ V2 is also a subspace of W. V1 ^ V2 denotes the intersection of V1 and V2, i.e. the set of vectors which belong to both V1 and V2.

Please explain why it is true and if it is false give a counterexample.

Find the six nonisomorphic trees on 6 vertices, and
for each compute the number of distinct spanning trees in K6 isomorphic
to it.

You and your classmates are boarding a lifeboat from a ship sinking in the middle of the ocean. You each are guaranteed a seat. Unfortunately, there are 10 others seeking a seat, as well, with only three seats remaining. As a group, you must decide which three people from the following list you will bring in your lifeboat and why?

• Restaurant manager, 67, married with two kids
• Croatian first-year medical student, 23, speaks little English
• Single mother, 34, has 3 kids
• LGBT male accountant, 58, a professional triathlete
• High school grad, 18, going into the army
• Homeless person, 40, Harvard dropout
• Successful entrepreneur, 49, chronic medical problems
• Top-ranked Olympic swimmer from Brazil, 21, no English
• Billionaire's daughter, 6, certified genius
• Comedian of Latino descent, 38, internationally famous