Let T : P3(R) → P4(R) be defined by T(f(x)) = 5f′(x)-∫ f(t)dt (integral from 0 to x)

1. Show that T is a linear transformation.

2.Find dim (P3(R)) and dim (P4(R)).

3.Find rank(T). Find nullity(T)
4. Is T one-to-one? Is T onto? Justify your answers.

Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker P is W^perp.

1. Find all solutions to the following linear congruences using Fermat’s Little Theorem or Euler’s Theorem to help you. Show all your steps.

(a) 3462x ≡ 6 173 (mod 59)

(b) 27145x ≡ 1 (mod 42)

D^2 (D + 1)y(t)= (D^2 +2)f(t)

a.) Find the characteristic polynomial, characteristic equation, characteristic roots, and characteristic modes of the system.

b.) Find y_o(t), the zero-input component of response y(t) for t>=0, if the the initial conditions are   y_0 (0) = 4, y_0' (0) = 3, and y_0'' (0) = -1

Let f : [0, 1] → R and suppose that, for all finite subsets of [0, 1], 0 ≤ x1 < x2 < · · · < xn ≤ 1,

we have |f(x1) + f(x2) + · · · + f(xn)| ≤ 1. Let S := {x ∈ [0, 1] : f(x) ̸= 0}. Show that S is countable

Number Systems Choose any system of numeration (Egyptian, Roman, Mayan, Chinese, Hindu-Arabic, Greek, Babylonian, etc.) and answer the following questions: Is it an additive, multiplicative, or ciphered system? Why? Is there a radix or base? What is it? Is there a schema or rule for combining the numerals to represent numbers? Briefly describe the rule(s).

explain interest rate risk or maturity price risk faced by short term and long term investors in bonds using an example

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove

(a). (a^m)(a^n)=a^(m+n)

(b). (a^m)^n=a^(mn)

y"-3y'+2y=4t-8 , y(0)=2 , y'(0)=7
y(t)=?

Write one a MATLAB function that implements the Bisection method, Newton’s method and Secant Method (all in one function). Your function must have the following signature

function output = solve(f,options)

% your code here

end

where

the input is

• • f: the function in f(x) =0.
  • options: is a struct type with the following fields o method: bisection, newton or secant
  • tol: the tolerance for stopping the iterations.
  • maximum_iterations: the maximum number of iterations allowed.
  • initial_guess: that is P_0; if the method needs it
  • df: the derivative of f if the method needs it
  • initial_interval: if the method needs it

the output is also a struct type with the following fields

• • message: either ‘success’ or an error message.
• • root: the solution in case of success.
• • iterations: an array that saves all iterations of the algorithm. Each row represents an iteration of the algorithm. Each row must contain P_n, f(P_n) and |P_n-P_n-1|.
• Write a script file that tests your function using the following equations

                                  600x^4 – 550x^3 +200x^2 – 20x -1 = 0

A) Find Eigen values and Eigen vectors for the matrix below.

A = ( 2 3 ; 1 5 ) this is a 2x2 matrix with 2 3 on the first row and 1 5 on the second row

(B) Write down the spectral decomposition of the matrix A.

(C) Is the matrix A positive definite matrix? Why?

5. Determine if the following sets along with the given operations form groups. If so, determine the identity element and whether or not they are Abelian. If not, explain why.

(a) GL(n, Z) where ∗ is matrix multiplication. This is the collection of all n × n nonsingular matrices with integral entries.

(b) Sym(X) where X is a nonempty set and f ∈ Sym(X) if and only if f : X → X is bijective where ∗ is composition.

(c) Aff(1, R), where Aff(1, R) := {fa,b : R → R : fa,b(x) = ax + b, a, b ∈ R, a 6= 0} and ∗ is composition. These are called the one-dimensional affine functions. What happens if we allow a = 0?

(d) T := {z ∈ C : |z| = 1} where ∗ is complex multiplication. We will again encounter T in later sections.

(e) SL(2, Z) where A ∈ SL(2, Z) if and only if A is a 2 × 2 matrix of integers for which det A = 1. What about SL(n, Z) where n ∈ N?