Decide, with proof, which of the following form a basis for a topologyon R, stating clearly any result you use.• B₁ = {[-a, a] : a € R, a > 0}• B₂ = {[a, a + 1] : a € Z}● B3 = {[a, a + 1) : a € Z}

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Answered: Decide, with proof, which of the…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.3: Orthonormal Bases:gram-schmidt Process

Problem 17E: Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner...

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Exercise are stated using the notation in the last paragraph of Example. Prove or disprove for all . Example : In this example we outline the development of the quaternions as the set , For each in , we imitate conjugates in and write , It is left as exercise to verify that If , then , and . Thus has all the field properties except commutative multiplication.
Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..
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Prove that if m0 and (a,b) exists, then (ma,mb)=m(a,b).

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Decide, with proof, which of the following form a basis for a topology on R, stating clearly any result you use. • B₁ = {[−a, a] : a € R, a > 0} • B₂ = {[a, a + 1]: a € Z} • B3 = {[a, a + 1): a € Z}