Given an example of a set S with two binary operations such that S is closed under those binary operations but is not a vector space with respect to them.

Answer 1

Solution:

Let V = the set of positive real numbers.

Define the addition + as the usual addition of numbers.

Define the scalar multiplication as:

c • x = (2^c)(x) for all x ∈ V and c ∈ R (the real number field)

Clearly, the addition + is closed. Since 2^c > 0, the scalar multiplication • is also closed.

But 0 ∉ V, there is no zero vector for the addition, so V is not a vector space over R

with respect to + and •