Question
For each integer n greater or equal to 1. let Q be the set of all n-tuples of rational numbers. For example, Q^3 = (Vector(a,b,c) : a,b,c are elements of Q). Then the statement "Q^n" is a vector over the field of rational numbers "Q" is true for a) some but not all integers n greater or equal to 1 b) all integers n greater or equal to 1 c) no integer n greater or equal to 1
Answer
4.7
(1 Votes)
Ian
Advanced · Tutor for 1 years
Answer
The statement "Q^n is a vector space over the field of rational numbers Q" is referring to the concept of a vector space in linear algebra. A vector space is a set of vectors that can be added together and multiplied by scalars, following certain rules. In this case, Q^n refers to the set of all n-tuples of rational numbers. An n-tuple is a sequence (or ordered list) of n elements, which can be thought of as a kind of vector. The field of rational numbers, Q, is the set of all numbers that can be expressed as a ratio of two integers. The statement is asking whether Q^n is a vector space over Q, meaning whether the set of all n-tuples of rational numbers can be added together and multiplied by rational numbers in a way that follows the rules of a vector space.The answer is:**b) all integers n greater or equal to 1**This is because for any integer n greater or equal to 1, the set of all n-tuples of rational numbers can indeed be added together and multiplied by rational numbers in a way that follows the rules of a vector space. For example, if we take two n-tuples from Q^n, we can add them together by adding their corresponding elements. Similarly, we can multiply an n-tuple by a rational number by multiplying each of its elements by that number. These operations satisfy the properties required for a vector space, such as associativity, commutativity, and distributivity.