1.2 Span

What is span?

• The span is all possible linear combinations

  • we can write it as a vector contered on the cartisan plane and if we multiplyy it by a number that outtput 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 its 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)