Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, select all of the axioms that fail to hold. (Let u, v, and w be vectors in the vector space V, and let c and d be scalars.) R2, with the usual addition but scalar multiplication defined by [x]-[*] All of the axioms hold, so the given set is a vector space. 01. u + v is in V. 2.u+vV+u 3. (u + v) +w=u+ (v + w) 4. There exists an element 0 in V, called a zero vector, such that u + 0 = u. 5. For each u in V, there is an element -u in V such that u + (-u) 0. 6. cu is in V. 7. c(u + v) = cu + cv 8. (c + d)u cu + du 9. c(du) (cd)u 10. luu
Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, select all of the axioms that fail to hold. (Let u, v, and w be vectors in the vector space V, and let c and d be scalars.) R2, with the usual addition but scalar multiplication defined by [x]-[*] All of the axioms hold, so the given set is a vector space. 01. u + v is in V. 2.u+vV+u 3. (u + v) +w=u+ (v + w) 4. There exists an element 0 in V, called a zero vector, such that u + 0 = u. 5. For each u in V, there is an element -u in V such that u + (-u) 0. 6. cu is in V. 7. c(u + v) = cu + cv 8. (c + d)u cu + du 9. c(du) (cd)u 10. luu
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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![Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, select all of the axioms that fail hold. (Let u, v, and w be vectors in the vector space V, and let c and d be scalars.)
R2, with the usual addition but scalar multiplication defined by
{]-[*}
All of the axioms hold, so the given set is a vector space.
1. u + v is in V.
U2. u + V = V + U
3. (u + v) + w = u + (v + w)
4. There exists an element 0 in V, called a zero vector, such that u + 0 = u.
5. For each u in V, there is an element -u in V such that u + (-u) = 0.
6. cu is in V.
O 7. c(u + v) = cu + cv
8. (c + d)u = cu + du
9. c(du) = (cd)u
10. 1u = U
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Transcribed Image Text:Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, select all of the axioms that fail hold. (Let u, v, and w be vectors in the vector space V, and let c and d be scalars.)
R2, with the usual addition but scalar multiplication defined by
{]-[*}
All of the axioms hold, so the given set is a vector space.
1. u + v is in V.
U2. u + V = V + U
3. (u + v) + w = u + (v + w)
4. There exists an element 0 in V, called a zero vector, such that u + 0 = u.
5. For each u in V, there is an element -u in V such that u + (-u) = 0.
6. cu is in V.
O 7. c(u + v) = cu + cv
8. (c + d)u = cu + du
9. c(du) = (cd)u
10. 1u = U
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