1.2 Span

What is span?

• The span is all possible linear combinations

  • we can write it as a vector centered on the cartesian plane and if we multiply it by a number that output changes and we must find all possible outputs of that vector

• Example:

  • what is the span of (1,1), (2,2) is it a line or a plane?

    • we can see that it is a line because$s\left[\begin{array}{c}1\\ 1\end{array}\right]+t\left[\begin{array}{c}2\\ 2\end{array}\right]|s,t\in \mathbb{R}$
      • and if we turn it into a vector we can see that it’s just a line

  • What is the $span[(1,1),(1,0)]$ is it a line or a plane?$$\begin{align}

s\begin{bmatrix}
1 \
0
\end{bmatrix}+t\begin{bmatrix}
1 \
1
\end{bmatrix}|s,t\in\mathbb{R} \
\implies \begin{bmatrix}
x \
y
\end{bmatrix}=\begin{bmatrix}
s+t \
t
\end{bmatrix} \
\implies t=y \
\implies s=x-y
\end{align}$$
- $\therefore$all of ${\mathbb{R}}^2$ (plane)