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8. P {p(x) in P2: p(0)=0} 9. P {p(x) in P2: p" (0) 0} = 10. P = {p(x) in P₂: p(x) = p(-x) for all x}

8. P {p(x) in P2: p(0)=0} 9. P {p(x) in P2: p" (0) 0} = 10. P = {p(x) in P₂: p(x) = p(-x) for all x}

Advanced Engineering Mathematics
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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Linear algebra: please solve q8, 9 and 10 correctly and handwritten. Definition is also attached

DEFINITION 1 A set of elements V is said to be a vector space over a scalar field S if an addition operation is defined between any two elements of V and a scalar multiplication operation is defined between any element of S and any vector in V. Moreover, if u, v, and w are vectors in V, and if a and b are any two scalars, then these 10 properties must hold. Closure properties: (c1) u + vis a vector in V. (c2) av is a vector in V. Properties of addition: (al) u+v=v+u. (a2) u + (v + w) = (u + v) + w. (a3) There is a vector in V such that v + 0 = v for all v in V. (a4) Given a vector v in V, there is a vector -v in V such that v + (-v) = 0. Properties of scalar multiplication: (m1) a(bv) = (ab)v. (m2) a(u+v) = au + av. (m3) (a + b)v = av + bv. (m4) lv = v for all v in V.
Transcribed Image Text:DEFINITION 1 A set of elements V is said to be a vector space over a scalar field S if an addition operation is defined between any two elements of V and a scalar multiplication operation is defined between any element of S and any vector in V. Moreover, if u, v, and w are vectors in V, and if a and b are any two scalars, then these 10 properties must hold. Closure properties: (c1) u + vis a vector in V. (c2) av is a vector in V. Properties of addition: (al) u+v=v+u. (a2) u + (v + w) = (u + v) + w. (a3) There is a vector in V such that v + 0 = v for all v in V. (a4) Given a vector v in V, there is a vector -v in V such that v + (-v) = 0. Properties of scalar multiplication: (m1) a(bv) = (ab)v. (m2) a(u+v) = au + av. (m3) (a + b)v = av + bv. (m4) lv = v for all v in V.
In Exercises 6-11, the given set is a subset of a vector space. Which of these subsets are also vector spaces in their own right? To answer this question, determine whether the subset satisfies the 10 properties of Defini- tion 1. (Note: Because these sets are subsets of a vector space, properties (al), (a2), (m1), (m2), (m3), and (m4) are automatically satisfied.) 6. S = {v in R4: v₁ + v₁ = 0} 7. S = {v in R: V₁ + V4 = 1} 8. P = {p(x) in P₂: p(0) = 0} 9. P = {p(x) in P₂: p" (0) # 0} 10. P = {p(x) in P₂: p(x) = p(-x) for all x}
Transcribed Image Text:In Exercises 6-11, the given set is a subset of a vector space. Which of these subsets are also vector spaces in their own right? To answer this question, determine whether the subset satisfies the 10 properties of Defini- tion 1. (Note: Because these sets are subsets of a vector space, properties (al), (a2), (m1), (m2), (m3), and (m4) are automatically satisfied.) 6. S = {v in R4: v₁ + v₁ = 0} 7. S = {v in R: V₁ + V4 = 1} 8. P = {p(x) in P₂: p(0) = 0} 9. P = {p(x) in P₂: p" (0) # 0} 10. P = {p(x) in P₂: p(x) = p(-x) for all x}
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