Let V be a vector space over R with dim(V)= n≥3. Let S be a subspace of V with dim(S) = m. Prove the following.a) If m = n−1, then there are exactly two subspaces W of V such that S ⊆ W. (Don’t just identify two subspaces that contain S; you must prove why they are the only subspaces that contain S.) b) If 1 ≤ m < n−1, then there are infinitely many subspaces W of V such that S ⊆ W. 2. Let V be a vector space over R, and let S1 and S2 be subspaces of V.(a) Prove that Si C S2 = dim(S1) < dim(S2).(b) Prove that (Si C S2 and dim(S1)dim(S2)) = Si = S2.(c) Prove that if u1,..., uk are linearly independent vectors in V, and v e V withv ¢ span(u1,Uk), then u1,...,U½, V are linearly independent.3. Question 2 is meant to help you with the following...Let V be a vector space over R with dim(V) = n > 3. Let S be a subspace of V withdim(S)= m. Prove the following.(a) If m = n-1, then there are exactly two subspaces W of V such that S CW. (Don'tjust identify two subspaces that contain S; you must prove why they are the onlysubspaces that contain S.)(b) If 1 < m 2. Let V be a vector space over R, and let S1 and S2 be subspaces of V.(a) Prove that Si C S2 = dim(S1) < dim(S2).(b) Prove that (Sı C S2 and dim(S1) = dim(S2)) = S1 = S2.(c) Prove that if u1,..., uk are linearly independent vectors in V, and v e V withv 4 span(u1,Ug are linearly independent vectors in V, and v e V withU%), then u1,... , Uk, v are linearly independent.•.u

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Answered: Let V be a vector space over R with…

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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Let V be a vector space over R with dim(V)= n≥3. Let S be a subspace of V with dim(S) = m. Prove the following.

a) If m = n−1, then there are exactly two subspaces W of V such that S ⊆ W. (Don’t just identify two subspaces that contain S; you must prove why they are the only subspaces that contain S.)

b) If 1 ≤ m < n−1, then there are infinitely many subspaces W of V such that S ⊆ W.

2. Let V be a vector space over R, and let S1 and S2 be subspaces of V.
(a) Prove that Si C S2 = dim(S1) < dim(S2).
(b) Prove that (Si C S2 and dim(S1)
dim(S2)) = Si = S2.
(c) Prove that if u1,..., uk are linearly independent vectors in V, and v e V with
v ¢ span(u1,
Uk), then u1,...,
U½, V are linearly independent.
3. Question 2 is meant to help you with the following...
Let V be a vector space over R with dim(V) = n > 3. Let S be a subspace of V with
dim(S)
= m. Prove the following.
(a) If m = n-1, then there are exactly two subspaces W of V such that S CW. (Don't
just identify two subspaces that contain S; you must prove why they are the only
subspaces that contain S.)
(b) If 1 < m <n-1, then there are infinitely many subspaces W of V such that S CW.

2. Let V be a vector space over R, and let S1 and S2 be subspaces of V.
(a) Prove that Si C S2 = dim(S1) < dim(S2).
(b) Prove that (Sı C S2 and dim(S1) = dim(S2)) = S1 = S2.
(c) Prove that if u1,..., uk are linearly independent vectors in V, and v e V with
v 4 span(u1,
Ug are linearly independent vectors in V, and v e V with
U%), then u1,... , Uk, v are linearly independent.
•.
u

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