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 FigureĀ 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 \(\twovec{1}{0}\text{.}\)

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Figure3.5.6Lines through the origin form subspaces of \(\real^2\text{.}\)

In fact, any line through the origin forms a subspace, as seen on the right of FigureĀ 6. Indeed, any such line is the span of a nonzero vector on the line.

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