Let V be a non-empty set equipped with two operations named addition and scalar multiplication. Addition is a rule that assigns to any two elements u, v E V an element u + v. Scalar multiplication is a rule that assigns to any scalar k and any v e V an element kv. Such a set V is called a vector space if the following ten axioms are satisfied: 1. If u, v E V then u + v E V 2. u + v = v + u for all u, v E V 3. (u + v) + w = u + (v + w) for all u, v, w E V 4. There exists an element 0 E V such that u + 0 = u for all u V 5. For each u E V there exists an element -u e V such that u + (-u) = 0 6. If k is any scalar and u e V, then ku e V 7. k(u + v) = ku + kv for all u, v E V and all scalars k 8. (k + m)u = ku + mu for all u e V and all scalars k and m 9. k(mu) = (km)u for all u E V and all scalars k and m 10. lu = u for all u E V Now consider the set W = {(x, y) : x, y E R} with addition and scalar multiplication defined so that for all (x, y), (w, z) E W and k E R we have: (x, y) + (w, z) = (x + w, y + z) k(x, y) = (ky, kx) Which one of the following statements is true? O a. W is a vector space O b. W satisfies Axiom 8 but not Axiom 9 O c. W satisfies Axiom 9 but not Axiom 10 O d. W satisfies Axiom 6 but not Axiom 7 O e. W satisfies Axiom 7 but not Axiom 8
Let V be a non-empty set equipped with two operations named addition and scalar multiplication. Addition is a rule that assigns to any two elements u, v E V an element u + v. Scalar multiplication is a rule that assigns to any scalar k and any v e V an element kv. Such a set V is called a vector space if the following ten axioms are satisfied: 1. If u, v E V then u + v E V 2. u + v = v + u for all u, v E V 3. (u + v) + w = u + (v + w) for all u, v, w E V 4. There exists an element 0 E V such that u + 0 = u for all u V 5. For each u E V there exists an element -u e V such that u + (-u) = 0 6. If k is any scalar and u e V, then ku e V 7. k(u + v) = ku + kv for all u, v E V and all scalars k 8. (k + m)u = ku + mu for all u e V and all scalars k and m 9. k(mu) = (km)u for all u E V and all scalars k and m 10. lu = u for all u E V Now consider the set W = {(x, y) : x, y E R} with addition and scalar multiplication defined so that for all (x, y), (w, z) E W and k E R we have: (x, y) + (w, z) = (x + w, y + z) k(x, y) = (ky, kx) Which one of the following statements is true? O a. W is a vector space O b. W satisfies Axiom 8 but not Axiom 9 O c. W satisfies Axiom 9 but not Axiom 10 O d. W satisfies Axiom 6 but not Axiom 7 O e. W satisfies Axiom 7 but not Axiom 8
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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Transcribed Image Text:Let V be a non-empty set equipped with two operations named addition and scalar multiplication. Addition is a rule that assigns to any two
elements u, v E V an element u + v. Scalar multiplication is a rule that assigns to any scalar k and any v E V an element kv. Such a set V is
called a vector space if the following ten axioms are satisfied:
1. If u, v E V then u + v E V
2. u + v = v + u for all u, v E V
3. (u + v) + w = u + (v + w) for all u, v, w E V
4. There exists an element 0 E V such that u + 0 = u for all u E V
5. For each u E V there exists an element -u E V such that u + (-u) = 0
6. If k is any scalar and u E V, then ku E V
7. k(u + v) = ku + kv for all u, v E V and all scalars k
8. (k + m)u = ku + mu for all u E V and all scalars k and m
9. k(mu) = (km)u for all u E V and all scalars k and m
10. lu = u for all u E V
Now consider the set W =
{(x, y) : x, y ER} with addition and scalar multiplication defined so that for all (x, y), (w, z) E W and k E R we
have:
(x, y) + (w, z) = (x + w, y + z)
k(x, y) = (ky, kx)
Which one of the following statements is true?
а.
W is a vector space
O b. W satisfies Axiom 8 but not Axiom 9
C.
W satisfies Axiom 9 but not Axiom 10
O d. W satisfies Axiom 6 but not Axiom 7
е.
W satisfies Axiom 7 but not Axiom 8
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