> [!NOTE] Definition (Subspace of $\mathbb{R}^{n}$) > Let $n\in \mathbb{N}^{+} .$ Let $\mathbb{R}^{n}$ denote the [[Real n-Space|real n-space]]. A subspace of $\mathbb{R}^{n}$ is nonempty [[Subsets|subset]] $W\subset \mathbb{R}^{n}$ such that for all $\underline{v},\underline{w}\in W,$ and $\lambda\in \mathbb{R},$ we have $\underline{v}+\underline{w}\in W$ and $\lambda \underline{v}\in W.$ > [!Example] Example (Trivial subspaces of $\mathbb{R}^{n}$) > There are two particular subspaces that we refer to as trivial subspaces: $\{ \underline{0} \}$ and $\mathbb{R}^{n}$ where $\underline{0}$ denotes the [[Real Zero Vector|zero vector]].