Problem 1 (The positive quadrant of R2 is a vector space) In this problem, we will see how the defini- tion of the vector addition and scalar multiplication affects being a vector space or not. Define the positive quadrant Q of R2 as follows: Q = {(x, y) = R² | x > 0 and y > 0}. 1. Suppose we use the usual addition and multiplication of vectors, i.e., (x1,y1) + (x2, y2) = (x1 + x2, Y1 + y2) c(x, y) = (cx, cy). Does Qwith these operations define a vector space? If so, provide a proof; if not, justify why. 2. Now suppose we define vector addition and scalar multiplication in a different way: (x1,y1) + (x2, y2) = (x1x2, Y1Y2) c(x, y) = (x, y). Prove that Q with these modified operations form a vector space.
Problem 1 (The positive quadrant of R2 is a vector space) In this problem, we will see how the defini- tion of the vector addition and scalar multiplication affects being a vector space or not. Define the positive quadrant Q of R2 as follows: Q = {(x, y) = R² | x > 0 and y > 0}. 1. Suppose we use the usual addition and multiplication of vectors, i.e., (x1,y1) + (x2, y2) = (x1 + x2, Y1 + y2) c(x, y) = (cx, cy). Does Qwith these operations define a vector space? If so, provide a proof; if not, justify why. 2. Now suppose we define vector addition and scalar multiplication in a different way: (x1,y1) + (x2, y2) = (x1x2, Y1Y2) c(x, y) = (x, y). Prove that Q with these modified operations form a vector space.
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:Problem 1 (The positive quadrant of R2 is a vector space) In this problem, we will see how the defini-
tion of the vector addition and scalar multiplication affects being a vector space or not.
Define the positive quadrant Q of R2 as follows: Q = {(x, y) = R² | x > 0 and y > 0}.
1. Suppose we use the usual addition and multiplication of vectors, i.e.,
(x1,y1) + (x2, y2) = (x1 + x2, Y1 + y2)
c(x, y) = (cx, cy).
Does Qwith these operations define a vector space? If so, provide a proof; if not, justify why.
2. Now suppose we define vector addition and scalar multiplication in a different way:
(x1,y1) + (x2, y2) = (x1x2, Y1Y2)
c(x, y) = (x, y).
Prove that Q with these modified operations form a vector space.
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