4. Let R be endowed with its standard topology. Let A be a topological subspace of R.a) Is {3} open in A=[0,1)U{3} ?b) Are [0,1) and (0,1) open in A=[0,1]?c) Let neN. Is {n}open in A =N ?%3Dd) Show that [0,1] and (2,3)are both open in A=[0,1]U(2,3).e) What is the closure of | 0,in A=(0,1]?

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Answered: 4. Let R be endowed with its standard…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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)Consider 3= {(a. B.Y.6):6 = 2a - B and y = a}. Is a subspace for the vector space R* with respect to the operations addition and scalar multiplication in vector space? Explain properly. (ii) Let S {(1,1,0, 1), (1, -1,0,1), (0,1,2,1)}. Determine whether p = (2,3,2,3) belongs to %3D span S.
Determine if the followings statements are "True" or "False":- teyiniz.) • A subset H of a vector space V is a subspace of V if the zero vector is in H. vekteran ayıdır.) • R"-1 is a subspace of R" for n > 3. ( ydır.) a • Let H be the set of all 2 x 2 matrices of the form where a,b and c are arbitrary real numbers. H is a subspace of M2x2.( ular alt uzayıdır.) • A subspace is also a vector space.
2. a) b) Let V R³. Let W be a subset of V where W - -2 (0-3) 2 8 Show that W is a subspace of V. The vectors a = b { (+) ERODER} € R³, a, b € R 2a 3b EW. Show that 1 -2 ())+ (1) ₁ 2 8 E W
4. Let V = {ao+aj x+a2 x² | ao, a1, a2 E R} be a vector space of polynomial up to degree three, and p(x) = pPo+P1 x+p2 x², q(x) = qo+q1 x+q2 x² E V then show that = po.4o + pP1-q1 + P2-92 defines an inner product on V.
= {(1 a) Write the this subspace as a span. 35. Consider the subspace V a c) What is the dimension of V? 0 b b - c 2a C ) |a, b, c € R} ≤ M2x3 b) Show this spanning set is a basis for V by showing that it is linear independent.
3. Let V be a finite-dimensional inner product space, and suppose U is a subspace of V. Show that Pu₁ = I – PU.
Show that if W is a subspace of a finite-dimensional vec- tor space V, then W is finite-dimensional and dim W ≤ dim V.
A={1,2,3,4} & R={(1,1),(1,2),(2,1),(2,2,),(3,4),(4,3),(3,3),(4,4)} Determine Equivalence classes, Rank of R ; A/R.
Let V be an inner product space with dim V = 2, and suppose that L = {u+v, u+2v} is a linearly independent subset of V. Prove that if 1 is orthogonal to both u and v, then 1 = assume that V contains tuples. Also, be sure to use only results from our linear algebra course in your proof.) 0. (Note: You Imay not

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4. Let R be endowed with its standard topology. Let A be a topological subspace of R. a) Is {3} open in A=[0,1)U{3} ? b) Are [0,1) and (0,1) open in A=[0,1]? %3D c) Let neN. Is {n}open in A =N? d) Show that [0,1] and (2,3)are both open in A=[0,1]J(2,3). e) What is the closure of 0, in A=(0,1]?