Determine whether the set w is a subspace of R3 with the standard operations. If not, state why. (Select all that apply.) W = {(x1, 1/X1, X3): X1 and xy are real numbers, X1 + 0) W is a subspace of R W is not a subspace of R because it is not closed under addition. Wis not a subspace of R because it is not closed under scalar multiplication. X
(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...
EXPLAIN STEP BY STEP In Exercises 13 through 18 determine if the set of vectors S forms a subspace of the given vector space. Give reasons why S either is or is not a subspace. xn) in 13. S is the set of vectors of the form (x1, X2, ..., xn) in R”, with the x; real numbers and x2 = x4. 14. S is the set of vectors of the form (x1, X2, . R”, with the xị real...
a 0 0 where a b, and c are positive numbers. Let S be the unit ball whose bounding surface has the equation x-x R3 + R3 be a linear transformation determined by the matrix A= 1 Complete Let 0 b 0 + x 0 0 c parts a and b below. u1 x1 2 ,2 2 a Show that T S is bounded by the ellipsoid with the equation 1 Create a vector u = that is within set...
this is an optimization subject. that is example 2.33 Question 2 (6 Marks) (Chapter 2) Consider the function f : R3 -R defined as f(x1,2,3 +4eli++21), (G) Explain why f has a global minimum over the set Hint: Read Example 2.33 (i) Find the global minimum point and global minimum value of f over the set C. Example 2.33. Consider the function/(x1,x2)=xf+xỈ over the set The set C is not bounded, and thus the Weierstrass theorem does not guarantee the...
4. For this question, we define the following matrices: 1-2 0 To 61 C= 0 -1 2 , D= 3 1 . [3 24 L-2 -1] (a) For each of the following, state whether or not the expression can be evaluated. If it can be, evaluate it. If it cannot be, explain why. i. B? +D ii. AD iii. C + DB iv. CT-C (b) Find three distinct vectors X1, X2, X3 such that Bx; = 0 for i =...
2. Define a function g: R3 +R by g(x, y, z) = 2x2 + y2 + x2 + 2xz – 2y – 4. (a) Find all the critical points of g. (b) Compute the Hessian H, of g. (c) Classify the critical points of g. (d) The surface g(x, y, z) = 0 is an ellipsoid . Use the method of Lagrange multipliers to find the maximum value of the function (5 marks) (5 marks) (5 marks) f(x, y, z)...
co are 5. Suppose that the functions f :R3 R, g:R R, and h:RR ously differentiable and let (xo. o, zo) be a point in R3 at which f(xo, yo, zo-g(xo, yo, zo)sh(xo, yo, zo)s0 and By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a neighborhood of the point (xo, yo, Zo) the system of equations f(x, y, z) g(x, y, 2)0 hCx, y,...
Hi, could you post solutions to the following questions. Thanks. 2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
(a) This exercise will give an example of a connected space which is not locally connected. In the plane R2 , let X0 = [0, 1] × {0}, Y0 = {0} × [0, 1], and for each n ∈ N, let Yn = {1/n} × [0,1]. Let Y = X0 ∪ (S∞ n=0 Yn). as a subspace of R 2 with its usual topology. Prove that Y is connected but not locally connected. (Note that this example also shows that...