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For each of the sets below, determine if the set is a vector space or subspace. a. H = {],a, b real} b. V = 21 = {[; ], a, b real} С.

For each of the sets below, determine if the set is a vector space or subspace. a. H = {],a, b real} b. V = 21 = {[; ], a, b real} С.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.3: Vectors
Problem 14E
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For each of the sets below, determine if the set is a vector space or subspace. a. H = {2], a, b real} а. b. V = , а — b {E 1.a, b real} С. T= 0 b. C = {the set of all complex numbers of the form a + bi, where a, b are real} e. J= {the set of all polynomials such that p(t)divides evenly by (t – 1)} [Hint: write p(t) in factored form, with the factor (t – 1) pulled out. What does the other factor look like?] d. - {the set of all odd functions: f (-x) = -f(x)} W is the set of all nxn matrices such that A² = A. h. Q is the set of all exponential functions i. f. g. S is the set of all n x n singular matrices
Transcribed Image Text:For each of the sets below, determine if the set is a vector space or subspace. a. H = {2], a, b real} а. b. V = , а — b {E 1.a, b real} С. T= 0 b. C = {the set of all complex numbers of the form a + bi, where a, b are real} e. J= {the set of all polynomials such that p(t)divides evenly by (t – 1)} [Hint: write p(t) in factored form, with the factor (t – 1) pulled out. What does the other factor look like?] d. - {the set of all odd functions: f (-x) = -f(x)} W is the set of all nxn matrices such that A² = A. h. Q is the set of all exponential functions i. f. g. S is the set of all n x n singular matrices
Expert Solution
Introduction

Note :

 ?as per our company guidelines we are supposed to answer only first 3 sub-parts. Kindly repost other parts in next question.

 

Strategy :

To check wheather a set of vectors form any vector space/subspace we have to verify the following conditions :- 

  • Closure under addition
  • Commutativity
  • Associativity
  • Additive identity
  • Additive inverse
  • Multiplicative identity
  • Distributivity

And if the set of vectors fails to have any one of these properties, we can directly conclude that it is not a vector space/subspace.

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