Let V be a vector space over R, and let S1 and S2 be subspaces of V.a) Prove that S1 ⊆ S2 ⇒ dim(S1) ≤ dim(S2).b) Prove that (S1 ⊆ S2 and dim(S1) = dim(S2)) ⇒ S1 = S2c) Prove that if u1, . . . , uk are linearly independent vectors in V , and v v not ∈ span(u1, . . . , uk), 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 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

I have proved the all statements by taking a basis of vector spaces

Answered: Let V be a vector space over R, and let…

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, and let S₁ and S₂ be subspaces of V.

a) Prove that S₁ ⊆ S₂ ⇒ dim(S₁) ≤ dim(S₂).

b) Prove that (S1 ⊆ S2 and dim(S₁) = dim(S₂)) ⇒ S₁ = S₂

c) Prove that if u1, . . . , uk are linearly independent vectors in V , and v v not ∈ span(u1, . . . , uk), 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 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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