(a) Let V be the set of all ordered pairs (x, y) of real numbers, together with the nonstandard vector addition operation : V x V → V and standard scalar multiplication operation : Rx V → V given by: (x1, y₁) (x2, Y2) = (x₁ + x₂ + 1, y₁ + y2 + 1) k (x, y) = (kx, ky) Is V = (V,B,.) a vector space? Determine which of the vector space axioms V0-V4 and S0-S4 are satisfied, and which (if any) aren't.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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1 (a) Let V be the set of all ordered pairs (x, y) of real numbers, together with the nonstandard vector
addition operation : V x V → V and standard scalar multiplication operation : Rx VV given
by:
(x1, y₁) (x2, Y2) = (x₁ + x₂ + 1, y₁ + y2 + 1)
k (x, y) = (kx, ky)
Is V =
(V,B,.) a vector space? Determine which of the vector space axioms V0-V4 and S0-S4 are
satisfied, and which (if any) aren't.
(b) Let W be the set of real numbers of the form (x, 0) with the standard addition and scalar multiplication
operations. Is V a vector space? If not, why not?
Transcribed Image Text:1 (a) Let V be the set of all ordered pairs (x, y) of real numbers, together with the nonstandard vector addition operation : V x V → V and standard scalar multiplication operation : Rx VV given by: (x1, y₁) (x2, Y2) = (x₁ + x₂ + 1, y₁ + y2 + 1) k (x, y) = (kx, ky) Is V = (V,B,.) a vector space? Determine which of the vector space axioms V0-V4 and S0-S4 are satisfied, and which (if any) aren't. (b) Let W be the set of real numbers of the form (x, 0) with the standard addition and scalar multiplication operations. Is V a vector space? If not, why not?
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