> [!Definition] Definition (Standard Basis of $\mathbb{R}^n$) > Let $n\geq 1.$ The standard [[Basis of Vector Space|basis]] of the [[Real n-Space|real n-space]] $\mathbb{R}^{n}$ is the collection $\underline{e_{1}} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ \ldots \\ 0 \\ 0 \end{pmatrix} \; , \; \underline{e_{2}} = \begin{pmatrix} 0 \\ 1 \\ 0 \\ \ldots \\ 0 \\ 0 \end{pmatrix}, \dots, \underline{e_{n}} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ \ldots \\ 0 \\ 1 \end{pmatrix} \in \mathbb{R}^{n}$that is, the $\underline{e_{i}}$ is the vector whose components are all zero except the $i$th component which is 1. # Properties The standard basis is a [[Basis of Vector Space|basis]] of $\mathbb{R}^{n}$. The standard basis is an [[Orthonormal Subset of Euclidean Space|orthonormal set of vectors]].