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Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (x, y) + (w, z) = (xw, yz) where xw on the right-hand side denotes standard multiplication of the two real numbers x and w, and yz is interpreted similarly. Define scalar multiplication on W so that for all 1 E R and (x, y) E W we have 1(x, y) = (x², y^) where x on the right-hand side denotes the standard operation of raising a positive real number to the power of another real number, and similarly for y*. Recall that if V is a vector space, then Axiom 7 states that for all u, v E V and all 1 E R 2(u + v) = Au + Av and Axiom 10 states that for all v E V 1y = v Which one of the following statements is true? O a. W satisfies Axiom 10 but not Axiom 7. O b. W satisfies neither Axiom 7 nor Axiom 10. O c. W satisfies Axiom 7 but not Axiom 10. O d. W satisfies both Axiom 7 and Axiom 10.

Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (x, y) + (w, z) = (xw, yz) where xw on the right-hand side denotes standard multiplication of the two real numbers x and w, and yz is interpreted similarly. Define scalar multiplication on W so that for all 1 E R and (x, y) E W we have 1(x, y) = (x², y^) where x on the right-hand side denotes the standard operation of raising a positive real number to the power of another real number, and similarly for y*. Recall that if V is a vector space, then Axiom 7 states that for all u, v E V and all 1 E R 2(u + v) = Au + Av and Axiom 10 states that for all v E V 1y = v Which one of the following statements is true? O a. W satisfies Axiom 10 but not Axiom 7. O b. W satisfies neither Axiom 7 nor Axiom 10. O c. W satisfies Axiom 7 but not Axiom 10. O d. W satisfies both Axiom 7 and Axiom 10.

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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Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у) + (ш, z) %3D (хи, уz) where xw on the right-hand side denotes standard multiplication of the two real numbers x and w, and yz is interpreted similarly. Define scalar multiplication on W so that for all 1 E R and (x, y) E W we have 1(x, y) = (x², y') where x" on the right-hand side denotes the standard operation of raising a positive real number to the power of another real number, and similarly for y*. Recall that if V is a vector space, then Axiom 7 states that for all u, v E V and all 1 ER 1(u + v) = Âu + Âv and Axiom 10 states that for all v E V lv = v Which one of the following statements is true? О а. W satisfies Axiom 10 but not Axiom 7. O b. W satisfies neither Axiom 7 nor Axiom 10. O c. W satisfies Axiom 7 but not Axiom 10. O d. W satisfies both Axiom 7 and Axiom 10.
Transcribed Image Text:Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у) + (ш, z) %3D (хи, уz) where xw on the right-hand side denotes standard multiplication of the two real numbers x and w, and yz is interpreted similarly. Define scalar multiplication on W so that for all 1 E R and (x, y) E W we have 1(x, y) = (x², y') where x" on the right-hand side denotes the standard operation of raising a positive real number to the power of another real number, and similarly for y*. Recall that if V is a vector space, then Axiom 7 states that for all u, v E V and all 1 ER 1(u + v) = Âu + Âv and Axiom 10 states that for all v E V lv = v Which one of the following statements is true? О а. W satisfies Axiom 10 but not Axiom 7. O b. W satisfies neither Axiom 7 nor Axiom 10. O c. W satisfies Axiom 7 but not Axiom 10. O d. W satisfies both Axiom 7 and Axiom 10.
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