Problem 4. Let K be a field and let f ∈ K[x]. Show that if 1+f2 has a factor of odd degree in K[x] then there is an a ∈ K such that a2 = −1.
2 + 2 ) 2 16. + Problem 24. Show that: (a+b+c+d) (- [5 marks] Problem 25. Given any TEC (V) on an inner product space V define: [u, u] = (T(u),T(0) Is (u, v) (u, v) an inner product? If not, then provide conditions on T such that this becomes an inner product, and prove this completely. (5 marks Problem 26. Suppose TEC(V) and dim range T = k. Prove that I has at most k + 1 distinct...
a) Show that the problem of k-colorability is reduced to the problem of (k + 1)-colorability. b) We know that the problem of 2-colorability is in class P. Can we then deduce from question a) above that problem of 3-colorability is also in class P? Explain your answer.
(40 pts) 2a. Show that u(z) is the solution to the problem where k(x)-1 for x < 1/2 and k = 2 for x > 1 /2. 2b. Set up the weak form for the differential equation above and the resulting element stiffness and element load vector and calculate the element stiffness matrix and load vector for 4 quadratic elements by using the Gaussian quadrature that is going to exactly calculate the integrals Then set up the global K and...
2. Using a z-transform table, show that a) 2k+14[k – 1] +ek-[k] 2+ ane) b) kyku[k – 1] Hint: Express 1 k-1) in terms of u[k]. c) (2-cos(k)]u[k –1] 2-0.52+0.25 d) k(k-1) (k - 2)24-34[k - m e for m=0,1,2 or 3
Problem 2. We have 4 urns and for each k = 1, 2, 3, 4, um k contains k red and 10-k green balls. We choose an urn randomly and then draw a ball from it at random. What is the probabláty that we draw a red ball?
4. Show that the solution of optimization problem for K-means is not unique arg min
Problem 1: Summation of min-terms (2, 4, 6, 9, 11 12) Minimize with a K-map and then correct of any and all potential static Hazards. Show all work. Be neat!
4. Show that the sequence of iterates r(k) defined by r(k) 1 +(rk)2 converges for an arbitrary choice of x(0) є R. Carry out the first l0 itera- tions in Matlab, and comment on the order of convergence.
4. Show that the sequence of iterates r(k) defined by r(k) 1 +(rk)2 converges for an arbitrary choice of x(0) є R. Carry out the first l0 itera- tions in Matlab, and comment on the order of convergence.
Page 8 of 9 HW-04 Problem No. 4.7 /10 pts 6 2 k For what value(s) of k is y in the plane spanned by vi and v? Show all your work, do not skip steps Displaying only the answer is not enough to get credit Solution (Show all intermediate steps, formulas, calculations, explanations and comments below this line. Don't write above this line) 2 k -3 - 1 4 1 K
Page 8 of 9 HW-04 Problem No. 4.7...