Span, Basis, and Dimensions

Span

The span of a set, $U$, is another set, $V$, consisting of all the linear combinations of vectors in $U$. The set $U$ is referred to as a spanning set.

Examples

1. $span\bigg(\bigg\{\begin{bmatrix}1\\0\end{bmatrix},\begin{bmatrix}0\\1\end{bmatrix}\bigg\}\bigg)=\R^2$

We can create any vector in $\R^2$, say $\begin{bmatrix}\alpha & \beta\end{bmatrix}^T$, using a linear combination of the spanning set $X=\bigg\{\begin{bmatrix}1 & 0\end{bmatrix}^T, \begin{bmatrix}1 & 0\end{bmatrix}^T\bigg\}$. So, $X$ spans $\R^2$. That is:

$$\alpha\begin{bmatrix}1 \\ 0\end{bmatrix}+\beta\begin{bmatrix}0 \\ 1\end{bmatrix}=\begin{bmatrix}\alpha \\ \beta\end{bmatrix}$$

2. $span\bigg(\bigg\{\begin{bmatrix}a\\b\end{bmatrix}\bigg\}\bigg)=$