Question

Is it possible for a vector space to have exactly four elements -- i.e., v = {a, b, c, d}? if so, give an example of such a space.

A) Yes
B) No
C) Maybe
D) Occasionally

Answer 1

A vector space can indeed have exactly four elements, as seen in the set of two-dimensional vectors over the finite field GF(2), which contain four distinct vector combinations. Hence, the correct option is (A) Yes.

Step-by-step explanation:

It is possible for a vector space to have exactly four elements. An example of such a vector space is the set of two-dimensional vectors over the finite field GF(2), which contains only two elements: 0 and 1. Here, the vector space would consist of the following four vectors: {(0,0), (0,1), (1,0), (1,1)}.

This vector space is closed under vector addition and scalar multiplication defined in GF(2), fulfilling the necessary properties of a vector space. The addition of vectors is commutative as mentioned, fulfilling A + B = B + A, and vectors can indeed be added in any order.