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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 48E
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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.
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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