1. Let S C R³ be the subset S = {(0, y,0)" | y ER}. Prove that if 7, w E S and a E R, then & + w E Sand au E S. (This is what it means to prove that S is closed under addition and scalar multiplication.)2. Let V be a vector space over R. Prove that if a E R and a + 0, then a0 = 0.

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Answered: 1. Let SC R be the subset S = {(0,…

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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A vector space over R is a set V of objects (called vectors) together with two operations, addition and multiplication by scalars (real numbers), that satisfy the following 10 axioms. The axioms must hold for all vectors u, v, win V and for all scalars a, B in R.
Please answer the question in handwritten form
If the dimension of a vector space is n where n > 1 and v1 ∈V is a non-zero vector, then there exist vectors a2, · · · , an ∈ V such that {λv1,a2,··· ,an} spans V for all λ ∈ R. Is this statement true or false? Justify your answer.
Let u, v, w be vectors in a vector space V. Which of the following properties need not be true? B:u+ (v+w) = (v +u) + w A:u+v=v+u C: Every u EV has an additive inverse. E: Every u € V has a multiplicative inverse. G: If r ER then r(u+v+w) = ru+rv+rw D:v+ (u+w) = (v +u) + w F: There exists x EV such that x+u = u. H: There is exactly one additive identity element.
1. State if the following statement is true or false. The set of vectors B = {u= (1,1,4) ;v = (2,0,2);w = (0,2,1)} is linearly independent in the vector space R³. O True O False
please prove 3 and 4
3. Which of the following pairs of vector spaces are isomorphic? Justify your answers. F³ and P3 (F). F4 and P3(F). (a) (b) (c) (d) M2x2 (R) and P3(R). V = {A € M₂x2 (R) : tr(A) = 0} and R4.
Let u = (– 5, – 10, – 8), v = (– 5, 6, 5), and w = (1, – 10, – 1). Prove/disprove: The set of linear combinations of u, v, w, using scalar multiples from R , is a vector space. -
Let V be a vector space over R and let u, v, w € V be distinct vectors. Prove that {u, v, w} is linearly independent if and only if {u+v, u+w,v + w} is linearly independent.

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1. Let SC R be the subset S = {(0, y,0)" | y € R}. Prove that if ī, w e S and a E R, then i+ w ES and au E S. (This is what it means to prove that S is closed under addition and scalar multiplication.) 2. Let V be a vector space over R. Prove that if a E R and a 0, then að = 0.