Real Similarity Transformations

Standard basis vectors

Define $e_i \in \R ^n$ by $[e_i]_j = 1$ if $i = j$ and $0$ otherwise. Then $e_1, e_2, \dots , e_n \in \R ^n$ are called the standard basis vectors (canonical basis vectors) for $\R ^n$. For example, in $\R ^3$,

\[ e_1 = \bmat{1 \\ 0 \\ 0} \]