Bases and Dimensions

Note

Let U be a subspace of ${\mathbb{R}}^n$ and let B be a finite subset${}^3$ of U we say that B is a basis for U if

1. B is linearly independent
2. U = Span B

• Standard Basis - Let ${\vec{e}}_1,\dots ,{\vec{e}}_n\in {\mathbb{R}}^n$ be the columns of the n x n identity matrix I. The set {${\vec{e}}_1,\dots ,{\vec{e}}_n$} is a basis for ${\mathbb{R}}^n$, called the standard basis for ${\mathbb{R}}^n$
  • Standard basis for ${\mathbb{R}}^2$ is {${\vec{e}}_1,{\vec{e}}_2$} = {$\left[\begin{array}{c}1\\ 0\end{array}\right]$, $\left[\begin{array}{c}0\\ 1\end{array}\right]$ }
• If B = $\{{\vec{v}}_1,\dots ,{\vec{v}}_k\}$ is a basis for ${\mathbb{R}}^n$, then k = n, that is, every basis for ${\mathbb{R}}^n$ consists of exactly n vectors
• Let B = {${\vec{v}}_1,\dots ,{\vec{v}}_n$} be a set of n vectors in ${\mathbb{R}}^n$. Then B spans ${\mathbb{R}}^n$ if and only if B is linearly independent