Question

Determine if the statement is true or false, and justify your answer. A vector space V must have an infinite number of distinct elements.

a. True. If v₁, is in V, then cv₁, is in V for every real number , so V must have an infinite number of disti
b. False. Consider V = {0).
c. True, by a theorem.
d. True, by the definition of a vector space.
e. False. Consider V=R. eBook

Answer 1

A vector space does not necessarily have to have an infinite number of distinct elements. For example, the vector space V = {0} only has one element, which is the zero vector.

Step-by-step explanation:

The statement is false. A vector space does not necessarily have to have an infinite number of distinct elements. For example, the vector space V = {0} only has one element, which is the zero vector. The zero vector is an essential element in any vector space, and it represents the additive identity element.