###### Example3.5.5Subsets that are subspaces

Let's look in $$\real^2$$ and consider $$S\text{,}$$ the set of vectors lying on the $$x$$ axis; that is, vectors having the form $$\twovec{x}{0}\text{,}$$ as shown on the left of Figure 3.5.6. Any scalar multiple of a vector lying on the $$x$$ axis also lies on the $$x$$ axis. Also, any sum of vectors lying on the $$x$$ axis also lies on the $$x$$ axis. Therefore, $$S$$ is a subspace of $$\real^2\text{.}$$ Notice that $$S$$ is the span of the vector $$\twovec{1}{0}\text{.}$$

In fact, any line through the origin forms a subspace, as seen on the right of Figure 3.5.6. Indeed, any such line is the span of a nonzero vector on the line.

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