QUESTION 2. (a) Decide whether each of the following subsets of R’ is a subspace. Either...
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace of R3. Justify your answer. (b) For the subsets which are subspaces, find a basis and the dimension for each of them 2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace...
3. For each of the following sets, determine if it is a subspace of R3. If it is a subspace, prove it. If is is not a subspace provide an example showing how it violates at least one of the subspace axioms (a) B , y,z) E R3 (x, y, 2)l 1) (b) S (a b, 3b+ 2a,a-b) a, be R) [10 (c) P (7,5,8) s(1,-1,2)t (3, 1,4) s,te R)
Determine whether each of the following is a subspace of the relevant R". (a) V1 = {(x, y, z) | x, y ER, Z E Z} (b) V2 = {(2,4,4) + s(1, 2, 2) + t(4,5,7) | ste R} (c) V3 = {(a, b, c, d) | a, b, c, d e R, ab = 0}
4. (25 points) Which of the following subsets of R3 are subspaces.Explain. a) {(x, y, z) 1 x 0, y 0, z ? c) {(z, y, z) | x2 + y2 + z2-1} d) Is the set H of all matrices of the form |(a,0)T, (b,d)T] a subspace of the space of all 2x2 matrices with the usual matrix addition and scalar multiplica- tion?
For each of the following metric spaces (X, d) and subsets A S X decide whether A is open, closed, neither or both. You do not need to justify your answers. (a) X-Z, d is the discrete metric. ACX is any subset (b) X = R2, d = d2 is the Euclidean metric. A = {(x, yje R2 : x-y) (c) X = R2, d = d2 is the Euclidean metric. A (0, 1) × {0).
please answer to all parts 4. In each case, determine whether W is a subspace of the F-vector space V. (a) F=Q, V =R2, and W = Z?. (b) F=R, V = RP, and W = {(x,y) ER?:zy >0}. (c) F=R, V = R', and W = {(x,y,z) € R3: x > 0, y > 0, 2 >0}. (d) F=R, V = C, and W = {(x, y, z) ECS:x + 2y + iz=0}. (e) F = Q, V =...
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...
part a and b PROBLEM (HAND-IN ASSIGNMENT) Use the Subspace Test to determine whether the following sets W are subspaces of the given vector spaces: (A) The set W to be of all triples of real numbers (x, y, z) satisfying that 2x - 3y + 5z = 0 with the standard operations on Ris a subspace of R3. (B) The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication.
1. Prove that each of the following is a subspace. (a) W = {x: x = (x 1, 22, 23) and X1 + 12 = x;} (b) W = {p: p(t) = ata + b + c and a+b+c=0} (C) W = {A € R2x2 and A is upper triangular) (d) W = {f:f EC(0,1) and f(0 =0} 2. Show that the following subsets of A R2x2 are not subspaces. (a) W = {A : A is the singular matrix}...
I need help with these linear algebra problems. 1. Consider the following subsets of R3. Explain why each is is not a subspace. (a) The points in the xy-plane in the first quadrant. (b) All integer solutions to the equation x2 + y2 = z2 . (c) All points on the line x + z = 5. (d) All vectors where the three coordinates are the same in absolute value. 2. In each of the following, state whether it is...