Linear Algebra Questions and Answers

Let F be a finite field and |F| = pn, p a prime and n ∈ Z+. For each divisor m of n, prove that F has exactly one subfield E of F such that |E| = pm.

Find the Laurent series for 1/ z+z² on the indicated sets: (a) 0< z <1 (b) 1 < |z| (c) 0 <|z+1| < 1 (d) 1 < |z+1|

Use the Bisection method to find solutions accurate to within 10 for the following problem a. x-2-x=0 for 0≤x≤1 b. ex-x² + 3x-2=0 for 0≤x≤1 c. 2x cos(2x) - (x + 1)²=0 for-3 ≤x≤-2 and -1≤x≤0 d. x cos x -2x²+3x-1=0 for 0.2 ≤x≤0.3 and1.2<x< 1.3

Joel and Carina are buying a house with an area of 1700 square feet. They want to recarpet the entire house. How many square yards of carpet will they need to buy? 129.6 square yards 240.16 square yards 188.89 square yards 566.67 square yards

Find the general solution to the differential equation using undetermined coefficients. y"+2y=cos((sqrt 2)x)+xe^2x

A function f(x) is defined by f(x) = 5 for 0 ≤ x ≤ π. (a) Find the half range sine series expansion of the function f(x). (8 marks) (b) Sketch the graph of the odd extension of the function f(x) for -3π ≤ x ≤ 4π. (5 marks)

Let G be a finite group and p be a prime number. Let H be a p-subgroup of G. Then, iG(H)=iNG(H)(H)(mod p), where iG(H) is the index of H in G.

Advanced Math using laplace tansform please Proceed to solve each of: y"-3y' + 4y=0 y(0) = 2 y'(0) = 3

Graph the equation. Let x = -3, -2, 1, 0, 1, 2, and 3. y = 1 Find the following y-values. Then choose the correct graph of the equation to the right.

Graph the given system of inequalities. x - 4y ≥ -8 2x 8y ≤ 0

State the intersection of the following system of equations. If there are no solutions, make sure you clearly indicate how you know and describe the scenario that is resulting in no solutions. π₁:3x + 2y + 5z = 4 π₂: 4x - 3y +z = -1

Estimate the minimum number of subintervals to approximate the value of a. the error estimate formula for the Trapezoidal Rule. b. the error estimate formula for Simpson's Rule. The minimum number of subintervals using the Trapezoidal Rule is 49⁰. (Round up to the nearest whole number.) The minimum number of subintervals using Simpson's Rule is 10 (Round up to the nearest even whole number.) 1 √√√5x+8 -4 dx with an error of magnitude less than 10 using ...

Let M = R³ be consider M as an R-module. Let X₁=(2, 0, 0), x₂ =(0, 1, 0) and x3= (0, 0, 3). Then, prove that M = <X₁>+x₂> <x₂>.

Let e₁ and e₂ be idempotents in a ring R and regard R as a left R-module. Prove that Re₁ + R(e₂ - e₂e₁)=Re₁ + Re₂

Let μ be the Lebesque measure on B(R). S= { (n² + 2)/ (n² + 1), n ∈Z }, A =[-1,1]; Then (1) μ(S) = 1 and μ(A ∩ Q) = 0. (2) μ(S) = 0 and μ(A ∩ Q) = 0. (3) μ(S) = 0 and μ(A ∩ Q) = 2 (4) μ(S) = 1 and μ(A ∩ Q)=2. (A) Option 1 (B) Option 2 (C) Option 3 (D) Option 4

Suppose {u, v, w} is a set of vectors in a vector space V. Which of the following statements is equivalent to "{u, v, w} is linearly independent"? I. None of the vectors u, v or w is a linear combination of the other vectors in {u, v, w}. II. None of the vectors u, v or w is a multiple of any other single vector in {u, v, w}. III. If a, b, c are scalars then au+by+cu=0 implies a = b = c=0. IV. If a b c = 0, then au + bv + cw = 0. mark (X) the correct answer: A I. and II. BI. and III. CI. and IV. D II. and III. E II. and IV. FIII. and IV.

Construct parametric equations describing the graph of the following equation. x= 4y + 6 If y = 2 + t, find the parametric equation for x.

Let FC KCL be fields. Let a EL be algebraic over K and K an algebraic extension of F. Then, a is algebraic over F.

A parabola has a vertex at (-5,22) and passes through the point (-6,18). Determine its equation in vertex form.

Prove that the following are equivalent to each other for any ideal I of a commutative ring R. 1. I = √I 2. I = √√J for some ideal J of R. 3. I is the intersection of a class of prime ideals of R.

For each of the following, sketch the graph of the indicated transformed function given the graph y=x² (9 marks) y=2(x-1)² Original points: (2,0), (-2,0), (0,2), (0, -2) b) y=(x-1)²-3 original points: (-2,-2), (0,1), (-4,1), (1,4), (-5,4)

Write the complex number in rectangular form. 14(cos 225° + i sin 225°) The complex number is.

Solve the quadratic equation and express all nonreal complex solutions in terms of i x2-12x+45=0

Use the Euclidean algorithm to find the following. GCD(290,609) and LCM(290,609) Note: "GCD" means greatest common divisor and "LCM" means least common multiple. GCD(290,609) = LCM(290,609) =

Let H be a group and let a € H. We define the following set: N = {g € H: ga = ag}. Prove that N is a subgroup of H.

Find the invariant factors of a linear operator T on the Q-vector space Q26 whose elementary divisor are (x-1)³, (x+4), (x+4)², (x-7)², (x-1)5, (x-1)3, (x-7)3, (x-1)4, (x+4)³

27. A rectangle has an area of x^3 + x² - 7x + 2 square meters and a width of x-2 meters. Find its length. x² + x - 7 meters x² + 3x - 1 meters x²+2x-3 meters x²- 3x + 5 meters

Which of the following is a polynomial function in factored form with zeros at 0, -3, and 4? f(x) = x(x+3)(x-4) f(x) = x(x - 3)(x + 4) f(x) = x³ - x² - 12x f(x) = x³ + x² - 12x

Determine whether the function f(x) = |x|+x² +0.001 is even, odd or neither. odd even neither

Determine the number of solutions of the right Spherical triangle 1. b = 42°18. B = 42°18' 2. a =25° A = 40⁰

The linear transformation T:R2 R3 given by T x = 1 3 y 2 a (x 3 9 y) It is NOT injective for a equal to:

Expand and simplify -3(2x - 1)(x + 4) -6x²-27x - 12 6x² +27x + 12 -27x + 12 -6x² - 21x + 12

Let f: R5 → R³ be a linear mapping. Find the correct statement. (a) f is injective. (b) If dim (Ker(f)) = 2, then f is surjective. (c) f is invertible. (d) If dim(Ker(f))= 3, then f is surjective.

Given A = {1, 2, 3, 4). Let S4 be the permutation group on A and H = {e, (1 2) (3 4), (1 3) (2 4), (14) (2 3)} be the subgroup of S4. S4= {H, (1 2) H, (1 3) H, (2 3) H, (123) H, (132) H} Determine whether S4/H is isomorphic to S3.

Solve the quadratic equation by using the quadratic formula. 6x² +13x=63 Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The solution is x = (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. There are no real roots.

Model the situation using a linear system. Define two variables and write the equations. You do not need to solve the system. Jada's job is to collect money from the pop machines. From one machine she collects a total of 76 dimes and quarters (think of their values). If the total value is $13, how many dimes and quarters are there?

For the statement, find the constant of variation and the variation equation. y varies directly as the cube of x; y = 49 when x = 7 Find the constant of variation k. k=(Type an integer or a simplified fraction.) Find the direct variation equation given y = 49 when x = 7. (Type an equation. Use integers or fractions for any numbers in the equation.)

In a survey of 400 seniors, x percent said that they plan on majoring in physics. One university has used this data to estimate the number of physics majors it expects for its entering class of 3,300 students. If the university expects 66 physics majors, what is the value of x?

If the unit of foreign currency is the peso, in which case is the real exchange rate 1.2? A. The U.S. price is $2, the foreign price is 5 pesos, and the exchange rate is 3 pesos per dollar. B. The U.S. price is $3, the foreign price is 18 pesos, and the exchange rate is 5 pesos per dollar. C. The U.S. price is $5, the foreign price is 12 pesos, and the exchange rate is 2 pesos per dollar. D. The U.S. price is $10, the foreign price is 3 pesos, and the exchange rate is 4 pesos per dollar.

Verify that x = -4, y = 4, z = -2 is a solution of this system of equations: {x+y+z= -1x -y +4z = -12 2x+y +z = -5 A) Not a solution B) Solution

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. a. x² + 2x + _____ = ( )² b. x² - 4x + _____ =( )² b. t² + 10t + _____ =( )² d. y² + y + _____ =( )²

For which value of t is the set V below a subspace of R³? V = {(x, y, z) | 5x + 15y = 16x + 3y + 19z = 15t+4} Enter your answer exactly, using fractions if necessary. t=

Solve using the elimination method. 5x + 3y = 19 x-6y=11

3. Consider the real Cartesian plane with the standard distance function, d = the square root of [(x₂ - X₁)² + (x₂ -y ₁)²]. Determine a corresponding coordinate system for the line / given by y = 2x + 3 (that is, a function f:/--> R such that for any two points P and Q on I, d(P, Q)= f(Q)-f(P)I.) Show all of your calculations.

1. If F is any collection of subsets of X, then prove that there exists a smallest a-algebra M such that FCM.

Which of the following holds true for the three points a= 3 b= 1 c= -3 1 0 -2 -1 1 5 The points are not on the same line but are on the same plane. All points are on the same line and the same plane. The points are on the same line but not on the same plane. The points are not on the same line nor on the same plane.

Write the given system of equations as a matrix equation and solve by using inverses. 5x₁ - 2x₂ = k₁ 3x1- x₂ = K₂ a. What are x₁ and x₂ when k₁=-1 and k₂ = 7? x₁ = __ x₂ = ____

Prove that for all integers a and b, if a|b then a² |3b²

(a) Write down the Lagrangian and the necessary Kuhn-Tucker conditions for the problem max 1/2x-y subject to x + e^-x ≤y.x>0

Suppose that the function p(x) the function f(x) with a approximates maximum error of ε over the interval [a, b]. Then what is the error for the approximation of the integral [a,b] p(x)dx for the integral [a,b] f (x)dx.