Show that, in n-dimensional space, any n + 1 vectors are linearly dependent.

HINT: Given n+1 vectors, where each vector has n components, write out the equations that determine whether these vectors are linearly dependent or not. Show that these equations constitute a system of n linear homogeneous equations with n + 1 unknowns. What do you know about the possible solutions to such a system of equations?

Use Classic Runge-Kutta method with h = 1 to solve the system y” - y’ - 6y = 0, y(0) = 2, y’(0) = 3 on [0,1]

Show that the eigenfunctions un(x) are orthogonal .

In Hilbert's Axioms, all of the axioms of connection are independent of each other

5) Let the function f : ℝ³ → ℝ³ be given by f(x, y, z) = (2x + 2y, 2y + 2z, z + x).

a) Prove that f is one to one and onto
b) Find the inverse of f, i.e., f⁻¹.

Consider a square matrix A such that Ker(A2 ) = Ker(A3 ). Is Ker(A3 ) = Ker(A4 ). Explain your reasoning.

use muller's method to find the roots of the equation f(x) = sin x - x/2 =0 near x=2

How do you recognize in which situations the idea of strong induction might be useful?

You have just started a new job and are thrilled to learn that your new employer offers a 401(k) retirement plan to its employees. Your annual salary is $40,000. Assume the IRS allows you to contribute up to $24,000 to your 401(k). You’ve decided to contribute 7% of your annual salary to the plan.

Questions:

1. How much more money would you need to contribute to meet the maximum allowable contribution set forth by the IRS?

2. The company offers you a $.50 match for each dollar that you contribute between 2 and 5 percent of your annual salary. How much is the company match based on your 7% contribution?

3. Is this a defined benefit plan or defined contribution plan? Why?

Find a general solution of the inhomogeneous equation y′′ + 2y′ + 5y = f(x) for
the following cases: (i) f(x) = 1 (ii) f(x) = x2 (iii) f(x) = e−x sin2x (iv) f(x) = e−x (v)
sin2x

COMPUTING LESLIE MATRIX

Example After one year, we have only 250 fishes left. And then 125 have reached their reproduction rate. If we set f₃ = 8, then we are back to n = (1000, 0, 0): We see that n1 = (0, 250, 0), n2 = (0, 0, 125), n3 = (1000, 0, 0)

Exercise Write down the Leslie matrix for the previous example and calculate for various choices of n the population vectors ni. What do you observe?

Exercise Show that you can find some n such that n⁺ = Ln = n. If n = (a, b, c) then

n⁺ = (8c, 0. 25a, 0. 5b). Then (a, b, c) = (8c, 0. 25a, 0. 5b) determines a unique stable distribution n amongst the age groups. n itself is unique up to a factor.

Exercise Now change f₃ = 8 to numbers smaller as well as larger than 8, say 6 and 10. Then calculate again for various choices of n the population vectors ni. Can you still find some n such that n⁺ = n?