Question

How do you know if a vector is closed under scalar multiplication?

Answer 1

If a set of vectors is closed under addition, it means that if you perform vector addition on any two vectors within that set, the result is another vector within the set. For instance, the set containing vectors of the form would be closed under vector addition.

Answer 2

Explanation:

A set of vectors is said to be closed under scalar multiplication if when any vector in the set is multiplied by any scalar, the result is still a vector in the same set.

In order to determine if a vector is closed under scalar multiplication, you can take any vector from the set and multiply it by any scalar. If the resulting vector is also in the same set, then the set is closed under scalar multiplication.

For example, consider the set of two-dimensional vectors with non-negative integer components:

{(0,0), (1,0), (0,1), (1,1), (2,1), (1,2), (2,2), (3,2), ...}

To check if this set is closed under scalar multiplication, we can take one of the vectors, say (1,1), and multiply it by a scalar, say 3:

3(1,1) = (3,3)

Since both components of (3,3) are non-negative integers, it is also in the set. Therefore, the set is closed under scalar multiplication.

Note that it is not sufficient to simply check that the scalar multiplication of one vector and one scalar is in the set. You need to check that this is true for any vector in the set and any scalar.