Exercise 5.7: Let T-VGR I V=Dor 7ey be the topology for R defined in problem #2...
5. Let R be equipped with the Fort topology TFp, defined by TF,-(U C R l p ¢ U or R-U is finite), with pER a point. (i) Is (R, (i) Is (R, TFp) path-connected? TF) connected? 5. Let R be equipped with the Fort topology TFp, defined by TF,-(U C R l p ¢ U or R-U is finite), with pER a point. (i) Is (R, (i) Is (R, TFp) path-connected? TF) connected?
Exercise 5.13 please Exercise 5.13: In the topological space (R, C) (where C is the half-open line topology from Theorem 2.18), let A-(-3, 0Ju[, 3). Which of the following sets are open in the CA-topology and how do you know? a. -2, 0 С. (-1,0]UII, 3) e. (2, 3) f. 2, 3) Theorem 2.18: Let C-(VSRI V- or V-R or V-(a, oo) for some aER) Then C is a topology for R, called the half-open line topology. Exercise 5.13: In...
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
Let V P2(R) and let T V-V be a linear transformation defined by T(p)-q, where (x)(r p (r Let B = {x, 1 + x2, 2x-1} be a basis of V. Compute [TIB,B, and deduce if it is eigenvectors basis of
a. 6. Let T: R* → P2(R)be defined as T 2) = (a - 2d) + (c + 3b)x + (a - 2c)x Ld] I Find a basis for the Ker(T). (3pts) b. Find a basis for the Range(T) (3pts) c. Determine whether T is one-to-one. (2pts) d. Determine whether T is onto. (2pts)
3. Let T : P2(R) → P2(R) be defined by T(f(x)) = f'(x). Find an element v ∈ P2(R) such that v, T v, T^2 v is a basis of generalized eigenvectors of T.
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
(6) Let T R R² be defined by T (a, az) = (a, -a2, a., 29, +a). Let ß be the standard basis for 1R² and v= {(1,1,0, (0, 1, 1), (2, 3,3)} Compute [7]} .
For the rest of this problem, let V be a subspace of R" and let T: R + R" be an orthogonal transformation such that T[V] = V1. (b) Prove that n is even and that dim V = dimV+ = (c) Prove that T[v+] = V. (d) Prove that there is a basis B of R" such that the B-matrix of T has block form (T) = [% ] where Qi and Q2 are orthogonal matrices,
(TOPOLOGY) Prove the following using the defintion: Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...