Describe the polynomials in the principal ideal generated by x in Cx explicitly: The characteristic of...
(Abstract Algebra-Ring Theory) In the quotient ring Z2[x]/(z6 + 1), verify that the ideal consisting of all multiples of g(x) = x4 +x2 + 1 contains all polynomials of the form a +baaz2 + ba3 + az4 (6,2) triple redundancy code bx (the corresponding codewords form the In the quotient ring Z2[x]/(z6 + 1), verify that the ideal consisting of all multiples of g(x) = x4 +x2 + 1 contains all polynomials of the form a +baaz2 + ba3 +...
ring over Q in countably many variables. Let I be the ideal of R generated by all polynomials -Pi where p; is the ith prime. Let RnQ1,2, 3, n] be the polyno- mial ring over Q in n variables. Let In be the ideal of Rn generated by all ] be the polynomial rin 9. Let R = QlX1,22.Zg, 2 polynomials -pi, where pi is the ith prime, for i1,.,n. . Show that each Rn/In is a field, and that...
6.1.15. Let I be the ideal of Z[x] of all polynomials with even constant terms. Show that the I is generated by x and 2. Proof. type here
6.2.9. Show that if R is a ring with identity, then the principal ideal gener ated by x ERis 6.2.9. Show that if R is a ring with identity, then the principal ideal gener ated by x ERis
Question 4: 4. Show that the following polynomials form a basis for P3 1 - x, 1-x2 1 +x _X 5. Show that the following matrices form a basis for M22 -8 1 0 3 12 -6 -4 2 _ 13. Find the coordinate vector of v relative to the basis S = {v1, V2, V3} for R3 (a) v (2, -1 3); vi = (1,0, 0), v2 = (2, 2, 0) Vз — (3, 3, 3) (b) v (5,...
Find the Taylor polynomials of order 0, 1, 2, and 3 generated by fat a. f(x)= Vx, a=4 The Taylor polynomial with order 0 is P(x)=
(x0,y0), (x1,y1) Using natural cubic spline, how can I get ax^3+bx^2+cx+d formula? (can a, b, c, d=0) Describe explicitly the natural cubic spline that interpolates a table with only two entries: 0 Give a formula for it. Here, to and t are the knots. Describe explicitly the natural cubic spline that interpolates a table with only two entries: 0 Give a formula for it. Here, to and t are the knots.
Let p(x) = x2 + 3x + 1 ∈ Z5[x] and let (p) ▹ Z5[x] be the principal ideal generated by p. Put K = Z5[x]/(p). For f(x)=x2−1∈ Z5[x] and g(x)=x2+1∈Z5[x] find a,b∈Z5 such that (f + (p))(g + (p)) = a + bx + (p) in K.
Describe the end behavior of given polynomials. Sketch the behavior of the branches 5. f(x) =-3x3 +7x2 - 2 an 6. f(x) = -5x6-2x3 + x 7. f(x) = 4x4 - 3x2 + 2 8. f(x) = 2x + 3x2 - X
6. Consider a state-space system x = Ax+ Bu, y = Cx for which the control input is defined as u- -Kx + r, with r(t) a reference input. This results in a closed-loop system x (A-BK)x(t)+ Br(t) = with matrices 2 -2 K=[k1 K2 For this type of controller, ki, k2 ER do not need to be restricted to positive numbers - any real number is fine (a) What is the characteristic equation of the closed-loop system, in terms...