examples of non empty subsets

Question

Our linear algebra prof gave us this problem and I am not quite sure how to go about starting or what it is asking for.

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

Answer 1

This is what closed means in this context:

A subset $U \subseteq \mathbb{R}^2$ is said to be closed under scalar multiplication if the following holds:

$$\text{For every } \mathbf{x} \in U \text{ and } c \in \mathbb{R},\ c\mathbf{x} \in U$$

So he wants you to find a subset $U$ satisfying the above property but is NOT a subspace.

In this context, a subset $V\subseteq \mathbb{R}^2 $ is a subspace if:

1. $\mathbf{0} \in V$

2. V is closed under scalar multiplication

3. If $\mathbf{x,y}\in V$, then $\mathbf{x}+\mathbf{y} \in V$

So you need to find a subset $U$ that satisfies #2 but fails to satisfy #1 or #3. Hint: Really it will be #3 as #2 implies #1 if $U$ is non-empty.

Answer 2

• The question is asking for a set $S$ of points in the plane $\mathbb{R}^2$. Examples of sets of points in $\mathbb{R}^2$ include: "the set of points on the unit circle", "two parallel lines", "sixteen disconnected points", "the x-axis", etc.
• The set $S$ is required to be closed under scalar multiplication, which means that if you take any multiple of a point in $S$, the result is still in $S$.
• The set $S$ should be nonempty, which just means that it has to have at least one point in it to avoid trivial examples.
• A linear subspace is a set of points in $\mathbb{R}^2$ which has certain properties. One of those properties is that it must be closed under scalar multiplication. A linear subspace must satisfy several other required properties, though (closure under addition, for example.).

• To find a set $S$ which is closed under scalar multiplication but which doesn't qualify as a linear subspace, it's enough to find a set of points which is closed under scalar multiplication but which is missing one of the other required properties.

• It may help to think visually: if a set $S$ of points in $\mathbb{R}^2$ is closed under scalar multiplication, it means that you can pick any point in the set, and draw the line passing through that point and the origin. Every point along that line must also be in $S$.