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Q9. Let W be a subspace of R". (a) Prove that w+ is a subspace of R". (b) Prove that if a vector v belongs to both W and W+, then v must be the zero vector.
(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...
Determine whether the subset S is a subspace of R" or not. If it is a subspace, explain why it is, either by checking that the three defining properties of a subspace are satisfied or by using a result from class (for insta that the span of vectors subspace which is not satisfied (e.g. specific vectors u and v are in S but iu ö is not in S), Studying examples 3.38, 3.39 and 3.40 in the textbook could be...
Problem 5. A subset A C R is an affine subspace of R" if there exists a vector bE R" and an underlying vector subspace W of R" such that (a) Describe all the affine subspaces of R2 which are not vector subspaces of R2 (b) Consider A E Rnx, bER" and the system of linear equations AT . Prove that: (i) if Ais consistent, then its solution set is an affine subspace of R" with underlying (ii) if At...
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?
(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...
A coil having L = 150 H, R = 200 ohms is connected in series with 100 ohms resistor. A 240-V dc source is connected to the circuit at t = 0. What is the voltage across the coil at t=0.5 sec.
Determine whether W = -{l. atb a - b : a,beR > is a subspace of R3. La + 3b +5]
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)
Suppose that {ū1, ... , ūk} is a basis for a subspace W of R" and that the vector Ū E span{ū1, ... , ūk}. Then û = Proj, Ū = ū. True O False Suppose that W is a subspace of R" and that the vector ŪER" .Then if û = Projű we have Ilu - Oll < 110 - ūll for all vectors ū EW . That is, <- is the vector in W that is closest to...