Example 3.2.4

It is worth pointing out that we first encountered a basis long ago when we considered the vectors in \(\real^3\text{:}\)

\begin{equation*} \evec_1 = \threevec{1}{0}{0}, \evec_2 = \threevec{0}{1}{0}, \evec_3 = \threevec{0}{0}{1}\text{.} \end{equation*}

We see that these vectors are, in fact, the columns of the \(3\times3\) identity matrix, which confirms that this set forms a basis.

More generally, the set of vectors \(\evec_1,\evec_2,\ldots,\evec_m\) forms a basis for \(\real^m\text{,}\) which we call the standard basis for \(\real^m\text{.}\)