2. Let V be a vector space over R, and let S1 and S2 be subspaces of V.(a) Prove that Si C S22 = dim(S1) < dim(S2).(b) Prove that (Sı C S2 and dim(S1) = dim(S2)) = Si = S2.(c) Prove that if u1,..., ug are linearly independent vectors in V, and v e V withv ¢ span(u1,.,..., Uk), then u1, ..., Uk, 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 SCW. (Don'tjust identify two subspaces that contain S; you must prove why they are the onlysubspaces that contain S.)-(b) If 1 < m

I have solved by assuming basis of the givem subspaces

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 € V with v g span(u1,. , Ug), then u1, Uk, v are linearly independent.

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 € V with v g span(u1,. , Ug), then u1, Uk, v are linearly independent.

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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Question

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

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