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I did some linear algebra exercise and did the following: Give an example of a nonempty subset $U$ of the xy-plane with the property that $U$ is closed with respect to scalar multiplication but $U$ is not a subspace.
I believe the set $U=\{(x,y): xy=0\}$ should provide such an example. Are there any better examples? More visual ones?
Your example is perfect. Visually, it represents the union of the $x$ and $y$ axes (which is a union of two lines through the origin that are not collinear). It fails to be a subspace because it isn't closed with respect to vector addition.
To see your example is the union of the $x$ and $y$ axes, observe that: $$ \begin{align*} U &= \{(x,y): xy=0\} \\ &= \{(x,y): x=0 \text{ or } y=0\} \\ &= \{(x,y): x=0\} \cup \{(x,y):y=0\} \\ &= (y\text{-axis}) \cup (x\text{-axis}) \end{align*}$$
Any finite (or countable!) set of lines through the origin, at least two of which are not collinear.
