### Linear Congruences - The Chinese Remainder Theorem

Find all solutions of each of the systems of congruences:- (a) x is congruent to 1 (mod 2) (d) 4x is congruent to 2 (mod 6) x is congruent to 2 (mod 3)

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Find all solutions of each of the systems of congruences:- (a) x is congruent to 1 (mod 2) (d) 4x is congruent to 2 (mod 6) x is congruent to 2 (mod 3)

Z=sum of dots (uniform on {1,2,3,4,5,6} a. M_z(s)=? b. M_z(s)=M_x(s)^2 why? M_z stands for the mgf of x=sum of dots when two dice are tossed

Once you obtain eigenvalues of 8.248, 7.661, and -3.909 for the matrix 1 5 -3 5 4 2 -3 2 7 what are the next steps to obtain the eigenvectors.

Find the Eigenvalues and Eigenvectors of the following matrix 1 5 -3 5 4 2 -3 2 7

X^2 - 2xy - 3y^2 = 1 x - 2y = 2

Let A be a 4x4 matrix with minimal polynomial m(t)=(t^2 + 1)(t^2 - 3). Find the rational canonical form for A if A is a matrix over (a) the rational field Q (b)the real field R, (c) the complex field C.

Let T be a linear operator on a finite dimensional vector space V. Suppose the minimal polynomial for T is of the form P^n where p is an irreducible polynomial over the scalar field. Show that there is a vector x in V such that the T-annihilator of x is p^n.

1. Let V be a vector space of odd dimension (greater than 1) over the real field R. Show that any linear operator on V has a proper invariant subspace other than {0}.

If a subset A of a metric space X has diameter less than epsilon, then it can be covered with one open ball of radius epsilon. Prove. (We must use direct definitions only for the proof).

Show that the operator L defined by L[y](x) = ∫_o^1(x-t)^2ydt is a linear operator

Consider the perturbed linear system x' = (A + eB(t))x, x is an element of R^n, where A is a constant matrix, B is a bounded continuous matrix valued function, and e is a small parameter. Assume that all eigenvalues of A have non-zero real part. 1) Show that the only bounded solution of the system is 0. 2) If A ha

Prove the statements that are true and give counterexample to disprove those that are false. For all integers n, if n is prime then (-1)^n=-1 (READ:...then (-1) to the n power equal -1)

3x+4y=-5 5x+6y=7 Please solve this system of equations using the addition method. Thank you for your time!

Jack invested $30 000 and received $2 300 in interest. Part of money was invested at 10% the remainer at 5%, how much was invested at each rate?

Write as a system of 2 equations in 2 unknowns. Solve each system by substitution and show each step. Perimeter of a rectangle : the length of a rectangular swimming pool is 15 feet longer than the width. If the perimeter is 82 feet, what are the length and width?

There are hens + rabbits. The heads = 50 the feet = 134. How many hens & how many rabbits ? Flies + spiders sum 42 heads and 276 feet. How many of each class? J received $1000 and bought 9 packs of whole milk & skim milk that totalled $960 - How many packs bought of each kind? A number is composed of two integers and its sum

Linear Independence of the Set of Vectors Linearly Independent Linearly dependent

Solve the following cryptogram doing the following steps: 1. frequency count 2. do you think it is monoalphabetic substitution, polyalphabetic subsitution, or transposition? 3. Is this a clear decision? 4. Solve based on the above info. DVOLL PULID ZIWGL ZDLIO WULFM WVWFM WVWFK LMULF IVHHV MGRZO SFNZM UIVVW LNHGS VUR

Find the solution, if it exists, to this system of linear equations: x + 2y - z = -4 3x + 7y - 6z = -21 x + 4y - 6z = -17 Find the solution, if it exists, to this system of linear equations: x - z = 2 2x - y = 4 x + y + z = 6 A cookie company makes three kinds of cookies, oatmeal raisin, cho

Is the function c(x)=(x^2+4)/x an hyperbola ? How would you rotate it 45 degrees in an anti clockwise direction ?

Dtermine whether or not the following matrix A= 5 0 2 0 5 0 2 0 5 is diagonalizable. If it is, then determine P'-1(P inverse)AP.

Please solve the following: Sec^2(X)csc^2(X)=sec^2(X)+csc^2(X) Make sure to show all the required steps and work.

Find the solution of the initial value problem: y'' + 2y' + 2y = 0, y(0) = 2, y'(0) = -3

Find the solution of the initial value problem: y'' - 4y' + 3y = 0, y(0) = 2, y'(0) = 3

If A is nonsingular, show that the characteristic values of A^(-1) are the reciprocals of A, and that A and A^(-1) have the same characteristic vectors.

Compute the Wronskian of the given set of functions, then determine whether the function is linearly dependent or linearly independent: x^2 - x, x^2 + x, x^2, all x

Show that the set of all elements of R^3 of the form (a + b, -a, 2b), where a and b are any real numbers, is a subspace of R^3. Show that the geometric interpretation of this subspace is a plane and find its equation.

Note: C means set containment (not proper set containment), |G : K| means index of subgroup K in G, and G # K means K is a normal subgroup of G question: Let K C H C G be groups, where K # G and |G : K| is finite. Show that |G/K : H/K| is also finite and that |G/K : H/K|=|G : H|

Please see the attached file for the fully formatted problems. What is the rank and signature of the quadratic form −x2+4y2−2z2+4xy+4xz?

Let Tn = (tij) denote the nxn matrix such that for each index i, tji = a and tij = b for j not = to i. Verify that Tn = (a-b)In + bEn where En is the nxn matrix of all 1's. Find the determinate and the eigenvalues of Tn.