I'm tasked with finding the characteristic equation, eigenvalues, and bases for the eigenspaces of this matrix

[1, -3, 3]

[3,-5,3]

[6,-6,4]

After working on the problem, I believe the characteristic equation is (λ + 2)^2(λ-4), giving eigenvalues of -2 and 4 (please correct me if I'm wrong). However, I'm lost when finding the bases for the eigenspace because I'm not sure the eigenvectors I get are linearly independent.

u''-2u'-8u=0 u(0)= α， u'(0)=2π

y''+9y'=cosπt, y(0)=0, y'(0)=1

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2
y′ = 2(t + 1)y, y(0) = 1, solution: y(t) = et2+2t

y′ = 5t4y, y(0) = 1, solution: y(t) = et5
y′ = t3/y2, y(0) = 1, solution: y(t) = (3t4/4 + 1)1/3

For the IVPs above, make a log-log plot of the error of Explicit Trapezoidal Rule at t = 1 as a function ofhwithh=0.1×2−k for0≤k≤5.

Prove that the composition of two quotient maps is a quotient map.

1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the characteristic polynomial of T. You may wish to use the basis of Mn×n(F) consisting of the matrices eij + eji, eij −eji and eii.

2.  Let A = (a b c d) (2 by 2 matrix) and let T : M2×2(F) → M2×2(F) be defined asT (B) = AB. Represent T as a 4×4 matrix using the ordered basis {e11,e21,e12,e22}, and use this matrix to prove that the characteristic polynomial of T is the square of the characteristic polynomial of A.

1. For each matrix A below compute the characteristic polynomial χA(t) and do a direct matrix computation to verify that χA(A) = 0.

(4 3

-1 1)    (2 1 -1 0 3 0 0 -1 2) (3*3 matrix)

2.  For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ.

A= (9 42 -30 -4 -25 20 -4 -28 23),λ = 1,3
A= (2 -27 18 0 -7 6 0 -9 8) , λ = −1,2

3. Construct a 2×2 matrix with eigenvectors(4 3) and (−3 −2) with eigen-values 2 and −3, respectively.

4. Let A be the 6*6 diagonal matrix below. For each eigenvalue, compute the multiplicity of λ as a root of the characteristic polynomial and compare it to the dimension of the eigenspace Eλ.

(x 0 0 0 0 0 0 x 0 0 0 0 0 0 y 0 0 0 0 0 0 x 0 0 0 0 0 0 z 0 0 0 0 0 0 x)

5.   Let A be an 3*3 upper triangular matrix with all diagonal elements equal, such as (3 4 -2 0 3 12 0 0 3)

Prove that A is diagonalizable if and only if A is a scalar times the identity matrix.

1.(4-x^2)y''+2y=0, x0=0

(a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation.

(b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner)

. (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions.

(d) If possible, find the general term in each solution.

Let f : Rn → R be a differentiable function. Suppose that a point x∗ is a local minimum of f along every line passes through x∗; that is, the function

g(α) = f(x^∗ + αd)

is minimized at α = 0 for all d ∈ R^n.

(i) Show that ∇f(x∗) = 0.

(ii) Show by example that x^∗ neen not be a local minimum of f. Hint: Consider the function of two variables

f(y, z) = (z − py^2)(z − qy^2),

where 0 < p < q.

Use Laplace Transforms to solve the following second-order differential equation:  

y"-3y'+4y=xe^2x where y'(0)=1 and y(0)=2

y'''-2y"-y'+2y=xe^{x^2+2x}

a) Find a general solution to the corresponding homogeneous equation, given that e^2x is one.

b) In the method of variation of parameters, find v₁, where v₁e^2x + v₂y₂ + v₃y₃ = y_p is a particular solution to the inhomogeneous equation. Use the method of variation of parameters

Please explain and show work, thanks!

Define f : Z × Z → Z by f(x, y) = x + 2y. (abstract algebra)

(a) Prove that f is a homomorphism.

(b) Find the kernel of f.

Farmer Pickles wants Bob to paint the circular fence which encloses his sunflower field. If the parametric equations x = 18 cos(θ) and y = 18 sin(θ) describe the base of the fence (in yards) and the height of the fence is given by the equation h(x, y) = 12 + (2x − y)/6, then how many gallons of paint with Bob need to complete the project. Assume that one gallon of paint covers three hundred square feet of fence

(3) Use the Hungarian Method to find the minimum possible cost for assigning the following jobs to the following four workers. The three jobs are new flooring, a new roof, and a new boiler. The costs for each person are as follows: (25 pts.)

Steve: 105 for Flooring, 321 for Roofing, 580 for Boiler.

Hoshi: 215 for Flooring, 300 for Roofing, 500 for Boiler.

Iqbal: 150 for Flooring, 315 for Roofing, 520 for Boiler.

Daphne: 240 for Flooring, 280 for Roofing, 497 for Boiler.

Recall that a 5-bit string is a bit strings of length 5, and a bit string of weight 3, say, is one with exactly three 1’s.

a. How many 5-bit strings are there?

b. How many 5-bit strings have weight 0?

c. How many 5-bit strings have weight 1?

d. How many 5-bit strings have weight 2?

e. How many 5-bit strings have weight 4?

f. How many 5-bit strings have weight 5?

g. How many 5-bit strings have weight 9?

Bob signs a note promising to pay Marie $3875 in 3 years at 9.5% compounded monthly. Then, 102 days before the note is due, Marie sells the note to a bank which discounts the note based on a bank discount rate of 18.5%. How much did the bank pay Marie for the note? $