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"Show that Q (rational numbers) is a vector space over R (real field). That is, a "vector" is now a coordinate pair of only rational numbers and the scalars a & b in properties (e), (f), (g) on pg 223 are real numbers.

"Show that Q (rational numbers) is a vector space over R (real field). That is, a "vector" is now a coordinate pair of only rational numbers and the scalars a & b in properties (e), (f), (g) on pg 223 are real numbers.

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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**Vector Space over a Real Field** Consider the set \( \mathbb{Q}^2 \) (rational numbers) as a vector space over \( \mathbb{R} \) (real field). Here, a "vector" is defined as a coordinate pair consisting exclusively of rational numbers. The scalars \( a \) and \( b \) in properties (e), (f), and (g) on page 223 are real numbers. This formulation implies a structure where vectors are rational and scalars are real, highlighting the interplay between different numerical fields within vector spaces.
Transcribed Image Text:**Vector Space over a Real Field** Consider the set \( \mathbb{Q}^2 \) (rational numbers) as a vector space over \( \mathbb{R} \) (real field). Here, a "vector" is defined as a coordinate pair consisting exclusively of rational numbers. The scalars \( a \) and \( b \) in properties (e), (f), and (g) on page 223 are real numbers. This formulation implies a structure where vectors are rational and scalars are real, highlighting the interplay between different numerical fields within vector spaces.
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