2. Find the basis for the solution space of the homogeneous system: a. X+2y = 0 2x+4y=0 b. 3x+2y+4z=0 2x+ y - 2 = 0 x +y +3z =0
Is W = {(x, y, z, w) | x − y = 2z + w & w − y = 2x + 3z} a subspace? Justify your answer. If it’s a subspace, find a basis for W and compute dim W.
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W (6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y...
Your problem is to find the optimal solution to the following linear programming model where X, Y and Z represent the amounts of products X, Y and Z to produce in order to minimize some cost. Min 4X + 2Y + 6Z s.t. 6X + 7Y + 10Z ≤ 80 (1) 2X + 4Y + 3Z ≤ 35 (2) 4X + 3Y + 4Z ≥ 30 (3) 3X + 2Y + 6Z ≥ 40 (4) X,Y,Z ≥...
[B] Let W be the subspace of M22 given in problem [A] . (B.1) Show that the following set forms a basis for W: S = -5 (B.2) Obtain the coordinate vector for A = 3 relative to S. That is, find (A)s. -8 Show work! [B] Let W be the subspace of M22 given in problem [A] . (B.1) Show that the following set forms a basis for W: S = -5 (B.2) Obtain the coordinate vector for A...
2. a) Find the dimension of the solution space of the homogeneous linear system (1 point) x-3y + z = 0 2x-6y + 2z = 0 2x + 4y-82=0 b) Find a basis for the solution space. (1 point)
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal. 12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
Exercise 1. Tangent plane (15 pts) Let (5) be the surface given by the following equation. x2+y2 = 1+z2 An equation of the tangent plane to (S) at A(1,2,2) is: a. 2x + 4y – 4z = 1 b. x + y - z=0 c. x + 2y – 2z = 1 d. x + y - z = 2 e. None of the above a. b. C. O d. e. Exercise 2. Directional derivative (6 pts + 9 pts)...
solving using Gaussian elimination method to solve the questions above 1. Solve the following systems of linear equations using Gaussian elimination method. a) 3z 9 x 5y 2z 2 1 3 *+2y = 3 b) x+y 0 2x y+3z3 x-2y -z 3 c) xy+z 9 2x +4y-3z= 1 3x+6y 5z= 0 e) 2xz w 5 y-w-1 3x z-w 0 4x +y+ 2z +w 9 2. How many gallons of each of a 60 % acid solution and an 80 %...
2. In each of the following, find conditions on a, b, and c (if any) such that the system has (i) no solution, (ii) a unique solution, and (iii) infinitely many solutions. (b) Ix-2y = 4 | 2x +y – z = a 2x +az= 6 2y +3z= b | 3x – 4y + 5z = b IX – cz = 1 [3]