Problem 3. Let 21, 22, 23, 41, 42, y3 be some real numbers, and let A=...
Q2) Let 21, 22, 23, 24, as be real numbers. Compute det ((a.) (1 :) ( ) ( ) ( .)) 1 1
9. (a) Let P1(21, 91, zı), P2 (22, 42, z2),P3 (13, 93, 23) be three non-collinear points in R, that is, three points which do not all lie on a straight line. Then the equations of the plane through these three points is: 2 y 21 1 = 0 22 Y2 22 1 2 1 2141 I3 Y3 23 1 Page 1 (b) Find the equation of the plane through P1(1,2,2), P2(1, 2, -1), P3(0,1,2)
3. (a) Let z1,z2, z3 € C, prove the following identity: (21 - 22)(22 – 23)(23 – £1) = (22 - 23)+23(23 – £1)+23(21 - 22). (b) In AABC, P is a point on the plane II containing A, B and C. Prove that aPA +bPB2 +cPC2 > abc.
DISCRETE MATHS 8. Let 띠,22, , z,, be positive real numbers, and let ~ = (z,a2 xn)Un be their geometric mean. (For example, the geometric mean of 3 and 27 is 9.) Prove that z S i for some k.
Problem 3. Let V and W be vector spaces, let T : V -> W be a linear transformation, and suppose U is a subspace of W (a) Recall that the inverse image of U under T is the set T-1 U] := {VE V : T(v) E U). Prove that T-[U] is a subspace of V (b) Show that U nim(T) is a subspace of W, and then without using the Rank-Nullity Theorem, prove that dim(T-1[U]) = dim(Unin (T))...
Assignment 1-2019 1. X: 23 22 44 12 25 24 16 50 42 21 21 36 23 22 41 Y: 50 50 50 50 50 49 43 44 48 47 46 8 3 4 2 P: 5 3 5 7 12 5 43 15 17 11 48 50 41 12 50 a) Make a frequency distributions for data set x and i=1 b) Combine set x, y and p and make a grouped frequency distribution (use i = 4, first bin 2-5). c) Using the same data and distribution add a column that shows the proportion frequency and the relative frequency (%) d) add a further column that shows the cumulative frequency. 2. Use the following grouped frequency distribution to answer the following questions. Score f...
Problem 3: Let f(x) be a function on the set of real numbers r > 1. Define the function g(x) for x by 1 g(s)-Σf(r/n). 1<nsr Prove that f(s) -Σμ(n)g (r/n). = 1nsz Here is the Möbius function
1 2 0 42 3 40 -80 64 48 -288 40 13 26 21-15 94-13) and 5 10 8-6 365 2 4 0 8 -6 -10 0 13 2-1 3·Let A = C 4 8 3 25 -2 9 5 1 42 3-1 9 10 22846-2 18 4 -4 3 21 3 2 334 15 26-2 14 5 48 -2 -10 -2 8 -8 814 16 28-23 148 36-6 56 (a) Find a basis for Nul (A)nNul (C) (b) Find...
n du Consider the data set below: 22 12 21 22 S = 19 45 20 30 37 34 44 23 26 33 70 54 40 48 23 17 22 56 38 17 21 16 33 41 84 43 27 65 34 19 100 34 48 29 54 16 43 33 8934 43 31 37 77 43 43 Problem 1 (a) Make a stem and leaf display of S. (b) Calculate the mean I = 2 (c) Calculate the 10%-trimmed...
Problem 8 Let P4 be the space of polynomials of degree less than 4 with real coefficients. Define L: PA + P4 by L(p(x)) = 5xp" (x) – (3x + 2)p" (x) + 7p'(x) a) (5 pts) Find the matrix representing L with respect to the standard basis S = = {1, 2, 22, 23} of P4. Explain how this can be used to prove directly that L is a linear transformation. b) (4 pts) Let S' {(4+ 3x), (2...