1a) For which operations are the polynomials closed (i.e. if the operands are polynomials, the result will also be polynomial) Multiple answers:You can select more than one option A Addition B Subtraction C Multiplication D Division E Exponentiation F Composition 1b) Which of the following "Millenium Problems" is now considered solved? A Yang–Mills and Mass Gap B Riemann Hypothesis C P vs NP Problem D Navier–Stokes Equation E Hodge Conjecture F Poincaré Conjecture G Birch and Swinnerton-Dyer Conjecture 1c) Which...
3. Recall Po(t) is the set of polynomials of order less than or equal to 2. Consider the the set of vec- tors in Pt) B = {t",( - 1)? (t+1)"} (a) Show B is a basis for P2(t). (b) If E = (1,4,7") is the standard basis, calculate the change of basis matrices Pe- and P3-8 (c) Given v = 21" - 56 +3, find its components in B
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...
The prime factorization of a positive integer n is p^3. Which of the following is true? Explain and show your answers. I. n cannot be even II. n has only one positive prime factor. III, n has exactly three distinct factors.
vi) Consider the following polynomials in the vector space of polynomials of degree 3 or less, P3. Pi(x) 12 +3r2 +a3 P2(x) 132 Pa(r) 1242 P4(z) = 1-r + 3r2 + 2r3 Which of the following statements are true and which are false? Explain your answer. a) The set {Pi, P2,P3} is a basis for P3. b) The set {Pi,P2, p3,P4,P5} İs a linearly independent set in P3.
vi) Consider the following polynomials in the vector space of polynomials of...
(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1)) (b) Let φο : Zp[2] Zp be the evaluation homomorphism at 0. Compute φο(zp-1-1) and φο((1-1)(1-2) . . . (z-(p-1))) (c) Use parts (a) and (b) to conclude that (-1)--1.
(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1))...
4. (10+10pts.) Consider the homogeneous system 21 +22+ (3 - 2a).x3 = 0 2x1 + 12 + 7.03 - 14 = 0 -22 + 20.73 +2.04 = 0 21 +22 + 4x3 = 0 where a is a real constant. a. Find the value of a for which the dimension of the solution space of the system is 1. b. Find a basis of the solution space of the system for the value of a found in part (a).
Which of the following would be the mean of this data set: 5, 28, 12, 56, 4, 2, 30, 21 18.24 19.75 17.06 16.50
Which of the following statements is true? The prime rate is often the same as the discount rate. The prime rate is higher than the federal funds rate. The federal funds rate is higher than the prime rate. The federal funds rate and the prime rate are often the same.
Which of the following statements best describes the lements in Group 8A (18) of the periodic table? a. They are all liquids under normal conditions. b. They exist naturally as single atoms. c. They form ions of variable charge. d. They exist as diatomic molecules.