Give an example of a nonempty subset U of R2 such that U is

Give an example of a nonempty subset U of R^2 such that U is closed under scalar multiplication, but V is not a subspace of R^2.

Solution

The example of a nonempty subset U of R² such that U is closed under scalar multiplication but is not a subspace o R² :

Consider the subset Z ² .

It is closed under addition;

however, it is not closed under scalar multiplication.

Consider the set U = {(n, 0) : n Z} (Z denotes the set of integers).

Let v and w A. Then there are integers n and m such that v = (n, 0) and w = (m, 0).

We have v + w = (n + m, 0) and n + m is an integer if n and m both are so the set is closed under addition so U is closed under addition.

Also, v = (m, 0) and m is an integer if m is, so U is closed under additive inverses. However, U is not closed under scalar multiplication.

Indeed, 1/ 2 · (1, 0) = ( 1/ 2 , 0)not belongs to U even though (1, 0) U.

Therefore, U has the desired properties

For example 2(1, 1) = ( 2, 2) / Z 2 .